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7.1.1: Figures in 1 and 2 Dimensions - Mathematics


Learning Objectives

  • Identify and define points, lines, line segments, rays, and planes.
  • Classify angles as acute, right, obtuse, or straight.

You use geometric terms in everyday language, often without thinking about it. For example, any time you say “walk along this line” or “watch out, this road quickly angles to the left” you are using geometric terms to make sense of the environment around you. You use these terms flexibly, and people generally know what you are talking about.

In the world of mathematics, each of these geometric terms has a specific definition. It is important to know these definitions, as well as how different figures are constructed, to become familiar with the language of geometry. Let’s start with a basic geometric figure: the plane.

A plane is a flat surface that continues forever (or, in mathematical terms, infinitely) in every direction. It has two dimensions: length and width.

You can visualize a plane by placing a piece of paper on a table. Now imagine that the piece of paper stays perfectly flat and extends as far as you can see in two directions, left-to-right and front-to-back. This gigantic piece of paper gives you a sense of what a geometric plane is like: it continues infinitely in two directions. (Unlike the piece of paper example, though, a geometric plane has no height.)

A plane can contain a number of geometric figures. The most basic geometric idea is a point, which has no dimensions. A point is simply a location on the plane. It is represented by a dot. Three points that don’t lie in a straight line will determine a plane.

The image below shows four points, labeled A, B, C, and D.

Two points on a plane determine a line. A line is a one-dimensional figure that is made up of an infinite number of individual points placed side by side. In geometry, all lines are assumed to be straight; if they bend, they are called a curve. A line continues infinitely in two directions.

Below is line ( A B) or, in geometric notation, ( overleftrightarrow{A B}). The arrows indicate that the line keeps going forever in the two directions. This line could also be called line ( BA). While the order of the points does not matter for a line, it is customary to name the two points in alphabetical order.

The image below shows the points ( A) and ( B) and the line ( overleftrightarrow{A B}).

Example

Name the line shown in red.

Solution

The red line goes through the points ( C) and ( F), so the line is ( overleftrightarrow{C F}).

( overleftrightarrow{C F})

There are two more figures to consider. The section between any two points on a line is called a line segment. A line segment can be very long, very short, or somewhere in between. The difference between a line and a line segment is that the line segment has two endpoints and a line goes on forever. A line segment is denoted by its two endpoints, as in ( overline{C D}).

A ray has one endpoint and goes on forever in one direction. Mathematicians name a ray with notation like ( overrightarrow{E F}), where point ( E) is the endpoint and ( F) is a point on the ray. When naming a ray, we always say the endpoint first. Note that ( overrightarrow{F E}) would have the endpoint at ( F), and continue through ( E), which is a different ray than ( overrightarrow{E F}), which would have an endpoint at ( E), and continue through ( F).

The term “ray” may be familiar because it is a common word in English. “Ray” is often used when talking about light. While a ray of light resembles the geometric term “ray,” it does not go on forever, and it has some width. A geometric ray has no width; only length.

Below is an image of ray ( E F) or ( overrightarrow{E F}). Notice that the end point is ( E).

Example

Identify each line and line segment in the picture below.

Solution

Two points define a line, and a line is denoted with arrows. There are two lines in this picture: ( overleftrightarrow{C E}) and ( overleftrightarrow{B G}).

A line segment is a section between two points. ( overline{D F}) is a line segment. But there are also two more line segments on the lines themselves: ( overline{C E}) and ( overline{B G}).

Lines: ( overleftrightarrow{C E}, overleftrightarrow{B G})

Line segments: ( overline{D F}, overline{C E}, overline{B G})

Example

Identify each point and ray in the picture below.

Solution

There are four points: ( A, B, C ext{ and } D).

There are also three rays, though only one may be obvious.

Ray ( overrightarrow{B C}) begins at point ( B) and goes through ( C). Two more rays exist on line ( overrightarrow{A D}): they are ( overrightarrow{D A}) and ( overrightarrow{A D}).

Points: ( A, B, C, D)

Rays: ( overrightarrow{B C}, overrightarrow{A D}, overrightarrow{D A})

Exercise

Which of the following is not represented in the image below?

  1. ( overleftrightarrow{B G})
  2. ( overrightarrow{B A})
  3. ( overline{D F})
  4. ( overrightarrow{A C})
Answer
  1. ( overleftrightarrow{B G})

    Incorrect. A line goes through points ( B) and ( G), so ( overleftrightarrow{B G}) is shown. ( overleftrightarrow{B A}), is not shown in this image.

  2. ( overrightarrow{B A})

    Correct. This image does not show any ray that begins at point ( B) and goes through point ( A).

  3. ( overline{D F})

    Incorrect. There is a line segment connecting points ( D) and ( F), so ( overleftrightarrow{D F}) is shown. ( overleftrightarrow{B A}), is not shown in this image.

  4. ( overrightarrow{A C})

    Incorrect. There is a ray beginning at point ( A) and going through point ( C), so ( overrightarrow{A C}) is shown. ( overrightarrow{B A}), is not shown in this image.

Lines, line segments, points, and rays are the building blocks of other figures. For example, two rays with a common endpoint make up an angle. The common endpoint of the angle is called the vertex.

The angle ( A B C) is shown below. This angle can also be called ( angle A B C), ( angle C B A), or simply ( angle B). When you are naming angles, be careful to include the vertex (here, point ( B) as the middle letter.

The image below shows a few angles on a plane. Notice that the label of each angle is written “point-vertex-point,” and the geometric notation is in the form ( angle A B C).

Sometimes angles are very narrow; sometimes they are very wide. When people talk about the “size” of an angle, they are referring to the arc between the two rays. The length of the rays has nothing to do with the size of the angle itself. Drawings of angles will often include an arc (as shown above) to help the reader identify the correct ‘side’ of the angle.

Think about an analog clock face. The minute and hour hands are both fixed at a point in the middle of the clock. As time passes, the hands rotate around the fixed point, making larger and smaller angles as they go. The length of the hands does not impact the angle that is made by the hands.

An angle is measured in degrees, represented by the degree symbol, which is a small circle at the upper right of a number. For example, a circle is defined as having 360o. (In skateboarding and basketball, “doing a 360" refers to jumping and doing one complete body rotation.)

A right angle is any degree that measures exactly 90o. This represents exactly one-quarter of the way around a circle. Rectangles contain exactly four right angles. A corner mark is often used to denote a right angle, as shown in right angle ( D C B) below.

Angles that are between 0o and 90o (smaller than right angles) are called acute angles. Angles that are between 90o and 180o (larger than right angles and less than 180o) are called obtuse angles. And an angle that measures exactly 180o is called a straight angle because it forms a straight line!

Example

Label each angle below as acute, right, or obtuse.

Solution

You can start by identifying any right angles.

( angle G F I) is a right angle, as indicated by the corner mark at vertex ( F).

Acute angles will be smaller than ( angle G F I) (or less than 90o). This means that ( angle D A B) and ( angle M L N) are acute.

( angle T Q S) is larger than

( angle G F I), so it is an obtuse angle.

( angle D A B) and ( angle M L N) are acute angles.

( angle G F I) is a right angle.

( angle T Q S) is an obtuse angle.

Example

Identify each point, ray, and angle in the figure below.

Solution

Begin by identifying each point in the figure. There are 4: ( E, F, G, ext { and } J).

Now find rays. A ray begins at one point, and then continues through another point towards infinity (indicated by an arrow).

Three rays start at point ( J): ( overrightarrow{J E}), ( overrightarrow{J F}), and ( overrightarrow{J G}). But also notice that a ray could start at point ( F) and go through ( J) and ( G), and another could start at point ( G) and go through ( J) and ( F). These rays can be represented by ( overrightarrow{G F}) and ( overrightarrow{F G})

Finally, look for angles.

( angle E J G) is obtuse, ( angle E J F) is acute, and ( angle F J G) is straight. (Don’t forget those straight angles!)

Points: ( E, F, G, J)

Rays: ( overrightarrow{J E}, overrightarrow{J G}, overrightarrow{J F}, overrightarrow{G F}, overrightarrow{F G})

Angles: ( angle E J G, angle E J F, angle F J G)

Exercise

Identify the acute angles in the image below.

  1. ( angle W A X, angle X A Y, ext { and } angle Y A Z)
  2. ( angle W A Y ext { and } angle Y A Z)
  3. ( angle W A X ext { and } angle Y A Z)
  4. ( angle W A Z ext { and } angle X A Y)
Answer
  1. ( angle W A X, angle X A Y, ext { and } angle Y A Z)

    Incorrect. ( angle W A X) and ( angle Y A Z) are both acute angles, but ( angle X A Y) is an obtuse angle. So only ( angle W A X) and ( angle Y A Z) are acute angles.

  2. ( angle W A Y ext { and } angle Y A Z)

    Incorrect. ( angle Y A Z) is an acute angle, but ( angle W A Y) is an obtuse angle. Both ( angle W A X) and ( angle Y A Z) are acute angles.

  3. ( angle W A X ext { and } angle Y A Z)

    Correct. Both ( angle W A X) and ( angle Y A Z) are acute angles.

  4. ( angle W A Z ext { and } angle X A Y)

    Incorrect. ( angle W A Z) is a straight angle, and ( angle X A Y) is an obtuse angle. Both ( angle W A X) and ( angle Y A Z) are acute angles.

Learning how to measure angles can help you become more comfortable identifying the difference between angle measurements. For instance, how is a 135o angle different from a 45o angle?

Measuring angles requires a protractor, which is a semi-circular tool containing 180 individual hash marks. Each hash mark represents 1o. (Think of it like this: a circle is 360o, so a semi-circle is 180o.) To use the protractor, do the following three steps:

  1. line up the vertex of the angle with the dot in the middle of the flat side (bottom) of the protractor,
  2. align one side of the angle with the line on the protractor that is at the zero degree mark, and
  3. look at the curved section of the protractor to read the measurement.

Supplemental Interactive Activity

For practice using a protractor, try out the activity below:

The example below shows you how to use a protractor to measure the size of an angle.

Example

Use a protractor to measure the angle shown below.

Solution

Use a protractor to measure the angle.

Align the blue dot on the protractor with the vertex of the angle you want to measure.

Rotate the protractor around the vertex of the angle until the side of the angle is aligned with the 0 degree mark of the protractor.

Read the measurement, in degrees, of the angle. Begin with the side of the angle that is aligned with the 0o mark of the protractor and count up from 0o. This angle measures 38o.

The angle measures 38o.

Exercise

What is the measurement of the angle shown below?

  1. 45o
  2. 135o
  3. 145o
  4. 180o
Answer
  1. 45o

    Incorrect. It looks like you started counting from the wrong side. In the picture above, notice how the bottom side of the angle is aligned with the 0o on the outside of the protractor. Continue to follow these numbers clockwise (10, 20, 30, ...) until you get to the point where the other side of the angle crosses the protractor. The correct answer is 135o.

  2. 135o

    Correct. This protractor is aligned correctly, and the correct measurement is 135o.

  3. 145o

    Incorrect. It looks like you thought the side of the angle crossed the protractor between 140o and 150o; it actually crosses between 130o and 140o. The correct answer is 135o.

  4. 180o

    Incorrect. You looked at the wrong side of the angle. The correct answer is 135o.

Geometric shapes and figures are all around us. A point is a zero-dimensional object that defines a specific location on a plane. A line is made up of an infinite number of points, all arranged next to each other in a straight pattern, and going on forever. A ray begins at one point and goes on towards infinity in one direction only. A plane can be described as a two-dimensional canvas that goes on forever.

When two rays share an endpoint, an angle is formed. Angles can be described as acute, right, obtuse, or straight, and are measured in degrees. You can use a protractor (a special math tool) to closely measure the size of any angle.


Two-dimensional space

Two-dimensional space (also known as 2D space, 2-space, or bi-dimensional space) is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point). The set ℝ 2 of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidean space. For a generalization of the concept, see dimension.

Two-dimensional space can be seen as a projection of the physical universe onto a plane. Usually, it is thought of as a Euclidean space and the two dimensions are called length and width.


How to construct a stem and leaf plot

We will now illustrate with three carefully chosen examples.

24, 10, 13, 2, 28, 34, 65, 67, 55, 34, 25, 59, 8, 39, 61

First, put this data in order

2, 6, 10, 13, 24, 25, 28, 34, 34, 39, 55, 59, 61, 65, 67

We will use 0, 1, 2, 3, 4, 5, and 6 as stems. The plot is displayed below:

A stem and leaf plot can help you quickly identify how frequently data occur. For example, a quick look at the figure above will show that the number 34 occurs most often. It can also help you identify quickly the least and the greatest data value.

This time, the data is already in order

104, 107, 112, 115, 115, 116, 123, 130, 134, 145, 147

We will use 10, 11, 12, 13, and 14 as stems. The plot is displayed below:

Sometimes, it is useful to show leafs on both sides of the stem. Say for instance you teach algebra in two different classes.

You may in this case want to compare performance for the classes to see which class performed better.

Grade for class A: 60, 68, 70, 75, 84, 86, 90, 91, 92, 94, 94, 96, 100, 100

Grade for class B: 60, 60, 70, 71, 73, 73, 75, 76, 77, 84, 85, 86, 91, 92

The plot is displayed below:

A quick look at the graph and you will see that class A performed a lot better than class B.

Class B has more scores in the 70s than class A.

Class A has more scores in the 90s than class B.

This is one of the good features of stem and leaf plots. It helps you to quickly look at the graph and examine the data to make sound conclusion.


HST Tutorial and Maths Formula

Half-square triangles, or HSTs, are one of the top quilt block units in quilting. The number of ways in how you use them is endless, and the number of HST quilt blocks surely number in the thousands! I would have to say that HST quilt blocks are my favourite for this very reason. You can make them any size, in any fabric and today I’m showing you how to make them using the two most popular methods – the traditional method and the alternative method. Both require two squares of fabric, but the size and method of sewing will determine how many HSTs you create in one go.

Traditional Method

This method will create 2 HST units from your two pieces of fabric. The benefit of this method is you’re working with the grain of the fabric, and therefore less chance of stretching when sewing the units together. This method is also really good for chain piecing, although it can be easy to lose track of which line you’re supposed to be sewing 1/4″ away from… Or is that just me?!

To determine what size squares you need to cut, all you need to know is what finished size you need – that is, what size you want them to be once all sewn together – and then add 7/8″ to that measurement. However, I would highly recommend adding the whole inch to leave yourself some excess to trim so as to improve your accuracy.

Step 1. Place two squares of fabric right side together. Draw a line diagonally from one corner to the opposite corner.

2. Sew 1/4″ from either side of the line backstitch at each end to secure your seams.

Step 3. Cut along the drawn line, open the fabric pieces and press the seams open.

You may need to trim down to even them all up or to achieve a specific size, depending on how accurate your cutting and sewing is!

Alternative Method

This method uses larger squares of fabric but you create 4 HST units from the fabric. It does create bias edges, but careful pressing (don’t “iron” back and forth, just press down with a hot iron and no steam) and pinning if you need that extra security should mean that you don’t get any bias stress.

The math for the alternative methods is a lot more complex, and instead we’re working with the unfinished measurements – that is, what size the HSTs are before you’ve sewn them all together. I’ve left the measurements “raw” so that you can work out how much excess you want to work with. For example, a 5″ charm square will give you four 3.18″ unfinished HSTs, which means you can trim them to 3″ square and end up with 2.5″ finished HSTs once all sewn up.

Step 1. Place two squares of fabric right side together. Sew around the outside a scant 1/4″ from the edge.

Step 2. Cut on both diagonals to get four pieces. Open the fabric pieces and press the seams open.

Step 3. Trim your HSTs the unfinished size needed for your project.

Compare The Pair

Which method you choose to use depends on how many HSTs you need, and how many of each combination you need! Let’s compare:

You need twenty-four 2.5″ unfinished HSTs for a 12″ finished Ripples block. Using the alternative method, you will need to cut six 4.5″ squares from both fabrics, which means you will need fabric that is about 9″ x 13.5″ for each fabric. And you will sew 6 seams.

Using the traditional method, you will need to cut twelve 3″ squares from both fabrics, which means you will need one 9″ x 12″ for each fabric. And you will sew 24 seams.

The larger the triangles, and the more you need to sew, the more fabric efficient the alternative method is, plus less seams to sew! If you are comfortable with the bias edge, it’s definitely the way to go. If you only need a few HSTs, like for a quilting bee block, or want a variety of prints within the block, the traditional method is the way to go to get the variety of prints needed and so as to not surprise your bee member with bias edges.

Other Tutorials

If you want to make eight HSTs at a time, make HSTs from strips of fabric, or want to check out triangle papers, then I have those tutorials too! Just follow those links to the one you’re interested in.


Area Worksheets

The links below take you to pages of printable area worksheets. Calculate the area of rectangles, squares, triangles, parallelograms, trapezoids, and circles.

Find the area of the shapes by counting the number of square unit tiles shown. These are very basic-level worksheets.

Find the areas of the rectangles and squares by using the formula area = length times width.

This page has a collection of worksheets for calculating the areas of triangles.

These PDFs have circles with the radius or diameter shown. Students must calculate the areas of the circles using the correct formula.

Here you'll find a series of worksheets on area of trapezoids and parallelograms.

These worksheets have irregular shapes (made of 2 or more rectangles rectilinear figures). Students find the areas of the individual rectangles and add them together.

Calculate the surface area of rectangular prisms, pyramids, cylinders, spheres, and irregular solid shapes.

This page includes worksheets for calculating the areas of circles. Also features circumference, radius, and diameter.

Find the perimeter of various polygons with these printable games, activities, and worksheets.


Unit Rates (Grade 7)



For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

Suggested Learning Targets

  • I can compute unit rates of quantities associated with ratios of fractions (length, area, & other quantities).
  • I can use proportional relationships to solve real-world problems.
  • I can simplify a rate, unit rate, ratio by dividing.
  • I can compute unit rate as a complex fraction.
  • Compute unit rates associated with ratios of fractions

Ratios & Unit Rates
Ratio is a comparison between two quantities by division
A rate is a ratio that compares quantities in different units
A unit rate is a rate with a denominator of 1.

Examples:
1. The table show prices for different packages of index cards. Which size has the lowest unit cost?

2. Convert 30gals/min cups/second. (Hint: 16 cups = 1 gallon)

4. Sam can make 15 pizzas in 2.5 hours. At this rate, how many could he make in an 8 hour work day? What is Sam's rate per hour?

5. Over 5 days, Kimberly sheared 300 sheep. How many would she be able to shear in 26 days? What was her rate per day?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.


11.3 Same Drawing, Different Scales

A rectangular parking lot is 120 feet long and 75 feet wide.

  • Lin made a scale drawing of the parking lot at a scale of 1 inch to 15 feet. The drawing she produced is 8 inches by 5 inches.
  • Diego made another scale drawing of the parking lot at a scale of 1 to 180. The drawing he produced is also 8 inches by 5 inches.
  1. Explain or show how each scale would produce an 8 inch by 5 inch drawing.
  2. Make another scale drawing of the same parking lot at a scale of 1 inch to 20 feet. Be prepared to explain your reasoning.
  3. Express the scale of 1 inch to 20 feet as a scale without units. Explain your reasoning.

Find the Median in Math

Problem: The Doran family has 5 children, aged 9, 12, 7, 16 and 13. What is the age of the middle child?

Solution: Ordering the childrens' ages from least to greatest, we get:

Answer: The age of the middle child is the middlemost number in the data set, which is 12.

In the problem above, we found the median of a set of 5 numbers.

Definition: The median of a set of data is the middlemost number in the set. The median is also the number that is halfway into the set. To find the median, the data should first be arranged in order from least to greatest.

To remember the definition of a median, just think of the median of a road, which is the middlemost part of the road. In the problem above, 12 is the median: it is the number that is halfway into the set. There are two children older than 12 and two children younger than 12. Let's look at some more examples.

Example 1: The Jameson family drove through 7 states on their summer vacation. Gasoline prices varied from state to state. What is the median gasoline price?

$1.79, $1.61, $1.96, $2.09, $1.84, $1,75, $2.11

Solution: Ordering the data from least to greatest, we get:

$1.61, $1.75, $1.79, $1.84, $1.96, $2.09, $2.11

Answer: The median gasoline price is $1.84. (There were 3 states with higher gasoline prices and 3 with lower prices.)

Example 2: During the first marking period, Nicole's math quiz scores were 90, 92, 93, 88, 95, 88, 97, 87, and 98. What was the median quiz score?

Solution: Ordering the data from least to greatest, we get:

87, 88, 88, 90, 92, 93, 95, 96, 98

Answer: The median quiz score was 92. (Four quiz scores were higher than 92 and four were lower.)

In each of the examples above, there is an odd number of items in each data set. In Example 1, there are 7 numbers in the data set in Example 2 there are 9 numbers. Let's look at some examples in which there is an even number of items in the data set.

Example 3: A marathon race was completed by 4 participants. What was the median race time?

Solution: Ordering the data from least to greatest, we get:

Since there is an even number of items in the data set, we compute the median by taking the mean of the two middlemost numbers.

Answer: The median race time was 4.3 hr.

Example 4: The salaries of 8 employees who work for a small company are listed below. What is the median salary?

$40,000, $29,000, $35,500, $31,000, $43,000, $30,000, $27,000, $32,000

Solution: Ordering the data from least to greatest, we get:

$27,000, $29,000, $30,000, $31,000, $32,000, $35,500, $40,000, $43,000

Since there is an even number of items in the data set, we compute the median by taking the mean of the two middlemost numbers.

Answer: The median salary is $31,500.

Summary: The median of a set of data is the middlemost number in the set. The median is also the number that is halfway into the set. To find the median, the data should be arranged in order from least to greatest. If there is an even number of items in the data set, then the median is found by taking the mean (average) of the two middlemost numbers.

Exercises

Directions: Find the median of each set of data. Click once in an ANSWER BOX and type in your answer then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.


Vector Operations

Let’s consider a vector v whose initial point is the origin in an xy - coordinate system and whose terminal point is . We say that the vector is in standard position and refer to it as a position vector. Note that the ordered pair defines the vector uniquely. Thus we can use to denote the vector. To emphasize that we are thinking of a vector and to avoid the confusion of notation with ordered - pair and interval notation, we generally write
v = .

The coordinate a is the scalar horizontal component of the vector, and the coordinate b is the scalar vertical component of the vector. By scalar, we mean a numerical quantity rather than a vector quantity. Thus, is considered to be the component form of v. Note that a and b are NOT vectors and should not be confused with the vector component definition.

Now consider with A = (x1, y1) and C = (x2, y2). Let’s see how to find the position vector equivalent to . As you can see in the figure below, the initial point A is relocated to the origin (0, 0). The coordinates of P are found by subtracting the coordinates of A from the coordinates of C. Thus, P = (x2 - x1, y2 - y1) and the position vector is .

It can be shown that and have the same magnitude and direction and are therefore equivalent. Thus, = = .

The component form of with A = (x1, y1) and C = (x2, y2) is
= .

Example 1 Find the component form of if C = (- 4, - 3) and F = (1, 5).

Note that vector is equivalent to position vector with as shown in the figure above.

Now that we know how to write vectors in component form, let’s restate some definitions.
The length of a vector v is easy to determine when the components of the vector are known. For v = , we have
|v| 2 = v 2 1 + v 2 2 Using the Pythagorean theorem
|v| = &radic v 2 1 + v 2 2 .

The length, or magnitude, of a vector v = is given by |v| = &radic v 2 1 + v 2 2 .

Two vectors are equivalent if they have the same magnitude and the same direction.

Let u = and v = . Then
= if and only if u1 = v1 and u2 = v2.

Operations on Vectors

To multiply a vector v by a positive real number, we multiply its length by the number. Its direction stays the same. When a vector v is multiplied by 2 for instance, its length is doubled and its direction is not changed. When a vector is multiplied by 1.6, its length is increased by 60% and its direction stays the same. To multiply a vector v by a negative real number, we multiply its length by the number and reverse its direction. When a vector is multiplied by 2, its length is doubled and its direction is reversed. Since real numbers work like scaling factors in vector multiplication, we call them scalars and the products kv are called scalar multiples of v.

For a real number k and a vector v = , the scalar product of k and v is
kv = k. = .
The vector kv is a scalar multiple of the vector v.

Example 2 Let u = and w = .Find - 7w, 3u, and - 1w.

Solution
- 7w = - 7. = ,
3u = 3. = ,
- 1w = - 1. = .

Now we can add two vectors using components. To add two vectors given in component form, we add the corresponding components. Let u = and v = . Then
u + v =

For example, if v = and w = , then
v + w = =

Before we define vector subtraction, we need to define - v. The opposite of v = , shown below, is
- v = (- 1).v = (- 1) =

Vector subtraction such as u - v involves subtracting corresponding components. We show this by rewriting u - v as u + (- v). If u = and v = , then
u - v = u + (- v) = + = =

We can illustrate vector subtraction with parallelograms, just as we did vector addition.

Vector Subtraction

It is interesting to compare the sum of two vectors with the difference of the same two vectors in the same parallelogram. The vectors u + v and u - v are the diagonals of the parallelogram.

Example 3 Do the following calculations, where u = and v = .
a) u + v
b) u - 6v
c)3u + 4v
d)|5v - 2u|

Solution
a) u + v = + = =
b)u - 6v = - 6. = - =
c) 3u + 4v = 3. + 4. = + =
d) |5v - 2u| = |5. - 2. | = | - | = | | = &radic (- 29) 2 + 21 2 = &radic 1282 &asymp 35,8

Before we state the properties of vector addition and scalar multiplication, we need to define another special vector—the zero vector. The vector whose initial and terminal points are both is the zero vector, denoted by O, or . Its magnitude is 0. In vector addition, the zero vector is the additive identity vector:
v + O = v. + =
Operations on vectors share many of the same properties as operations on real numbers.

Properties of Vector Addition and Scalar Multiplication

For all vectors u, v, and w, and for all scalars b and c:
1. u + v = v + u.
2. u + (v + w) = (u + v) + w.
3. v + O = v.
4 1.v = v 0.v = O.
5. v + (- v) = O.
6. b(cv) = (bc)v.
7. (b + c)v = bv + cv.
8. b(u + v) = bu + bv.

Unit Vectors

A vector of magnitude, or length, 1 is called a unit vector. The vector v = is a unit vector because
|v| = | | = &radic (- 3/5) 2 + (4/5) 2 = &radic 9/25 + 16/25 = &radic 25/25 = &radic 1 = 1.

Example 4 Find a unit vector that has the same direction as the vector w = .

Solution We first find the length of w:
|w| = &radic (- 3) 2 + 5 2 = &radic 34 . Thus we want a vector whose length is 1/&radic 34 of w and whose direction is the same as vector w. That vector is
u = w/&radic 34 = /&radic 34 = 34 , 5/&radic 34 >.
The vector u is a unit vector because
|u| = |w/&radic 34 | = = &radic 34/34 = &radic 1 = 1.

If v is a vector and v &ne O, then
(1/|v|)&bull v, or v/|v|,
is a unit vector in the direction of v.

Although unit vectors can have any direction, the unit vectors parallel to the x - and y - axes are particularly useful. They are defined as
i = and j = .

Any vector can be expressed as a linear combination of unit vectors i and j. For example, let v = . Then
v = = + = v1 + v2 = v1i + v2j.

Example 5 Express the vector r = as a linear combination of i and j.

Solution
r = = 2i + (- 6)j = 2i - 6j.

Example 6 Write the vector q = - i + 7j in component form.

Solutionq = - i + 7j = -1i + 7j =

Vector operations can also be performed when vectors are written as linear combinations of i and j.

Example 7 If a = 5i - 2j and b = -i + 8j, find 3a - b.

Solution
3a - b = 3(5i - 2j) - (- i + 8j) = 15i - 6j + i - 8j = 16i - 14j.

Direction Angles

The terminal point P of a unit vector in standard position is a point on the unit circle denoted by (cos&theta, sin&theta). Thus the unit vector can be expressed in component form,
u = ,
or as a linear combination of the unit vectors i and j,
u = (cos&theta)i + (sin&theta)j,
where the components of u are functions of the direction angle &theta measured counterclockwise from the x - axis to the vector. As &theta varies from 0 to 2&pi, the point P traces the circle x 2 + y 2 = 1. This takes in all possible directions for unit vectors so the equation u = (cos&theta)i + (sin&theta)j describes every possible unit vector in the plane.

Example 8 Calculate and sketch the unit vector u = (cos&theta)i + (sin&theta)j for &theta = 2&pi/3. Include the unit circle in your sketch.

Solution
u = (cos(2&pi/3))i + (sin(2&pi/3))j = (- 1/2)i + (&radic 3 /2)j

Let v = with direction angle &theta. Using the definition of the tangent function, we can determine the direction angle from the components of v:

Example 9 Determine the direction angle &theta of the vector w = - 4i - 3j.

Solution We know that
w = - 4i - 3j = .
Thus we have
tan&theta = (- 3)/(- 4) = 3/4 and &theta = tan - 1 (3/4).
Since w is in the third quadrant, we know that &theta is a third-quadrant angle. The reference angle is
tan - 1 (3/4) &asymp 37°, and &theta &asymp 180° + 37°, or 217°.

It is convenient for work with applied problems and in subsequent courses, such as calculus, to have a way to express a vector so that both its magnitude and its direction can be determined, or read, easily. Let v be a vector. Then v/|v| is a unit vector in the same direction as v. Thus we have
v/|v| = (cos&theta)i + (sin&theta)j
v = |v|[(cos&theta)i + (sin&theta)j] Multiplying by |v|
v = |v|(cos&theta)i + |v|(sin&theta)j.

Example 10 Airplane Speed and Direction. An airplane travels on a bearing of 100° at an airspeed of 190 km/h while a wind is blowing 48 km/h from 220°. Find the ground speed of the airplane and the direction of its track, or course, over the ground.

Solution We first make a drawing. The wind is represented by and the velocity vector of the airplane by . The resultant velocity vector is v, the sum of the two vectors:
v = + .

The bearing (measured from north) of the airspeed vector is 100°. Its direction angle (measured counterclockwise from the positive x - axis) is 350°. The bearing (measured from north) of the wind vector is 220°. Its direction angle (measured counterclockwise from the positive x - axis) is 50°. The magnitudes of and are 190 and 48, respectively.We have
= 190(cos350°)i + 190(sin350°)j, and
= 48(cos50°)i + 48(sin50°)j.
Thus,
v = +
= [190(cos350°)i + 190(sin350°)j] + [48(cos50°)i + 48(sin50°)j]
= [190(cos350°)i + 48(cos50°)i] + [190(sin350°)j + 48(sin50°)j]
&asymp 217,97i + 3,78j.
From this form, we can determine the ground speed and the course:
Ground speed &asymp &radic (217,97) 2 + (3,78) 2 &asymp 218 km/h.
We let &alpha be the direction angle of v. Then
tan&alpha = 3,78/217,97
&alpha = tan - 1 3,78/217,97 &asymp 1°.
Thus the course of the airplane (the direction from north) is 90° - 1°, or 89°.

Angle Between Vectors

When a vector is multiplied by a scalar, the result is a vector. When two vectors are added, the result is also a vector. Thus we might expect the product of two vectors to be a vector as well, but it is not. The dot product of two vectors is a real number, or scalar. This product is useful in finding the angle between two vectors and in determining whether two vectors are perpendicular.

The dot product of two vectors u = and v = is
u &bull v = u1.v1 + u2.v2
(Note that u1v1 + u2v2 is a scalar, not a vector.)

Example 11 Find the indicated dot product when
u = , v = and w = .
a)u &bull w
b)w &bull v

Solution
a) u &bull w = 2(- 3) + (- 5)1 = - 6 - 5 = - 11
b) w &bull v = (- 3)0 + 1(4) = 0 + 4 = 4.

The dot product can be used to find the angle between two vectors. The angle between two vectors is the smallest positive angle formed by the two directed line segments. Thus the angle &theta between u and v is the same angle as between v and u,and 0 &le &theta &le &pi.

If &theta is the angle between two nonzero vectors u and v, then
cos&theta = (u &bull v)/|u||v|.

Example 12 Find the angle between u = and v = .

Solution We begin by finding u &bull v, |u|, and |v|:
u &bull v = 3(- 4) + 7(2) = 2,
|u| = &radic 3 2 + 7 2 = &radic 58 , and
|v| = &radic (- 4) 2 + 2 2 = &radic 20 .
Then
cos&alpha = (u &bull v)/|u||v| = 2/&radic 58 .&radic 20
&alpha = cos - 1 (2/&radic 58 .&radic 20 )
&alpha &asymp 86,6°.

Forces in Equilibrium

When several forces act through the same point on an object, their vector sum must be O in order for a balance to occur. When a balance occurs, then the object is either stationary or moving in a straight line without acceleration. The fact that the vector sum must be O for a balance, and vice versa, allows us to solve many applied problems involving forces.

Example 13 Suspended Block. A 350-lb block is suspended by two cables, as shown at left. At point A, there are three forces acting: W, the block pulling down, and R and S, the two cables pulling upward and outward. Find the tension in each cable.

Solution We draw a force diagram with the initial points of each vector at the origin. For there to be a balance, the vector sum must be the vector O:

R + S + W = O.
We can express each vector in terms of its magnitude and its direction angle:
R = |R|[(cos125°)i + (sin125°)j],
S = |S|[(cos37°)i + (sin37°)j], and
W = |W|[(cos270°)i + (sin270°)j]
= 350(cos270°)i + 350(sin270°)j
= -350j cos270° = 0 sin270° = - 1.
Substituting for R, S, and W in R + S + W + O, we have
[|R|(cos125°) + |S|(cos37°)]i + [|R|(sin125°) + |S|(sin37°) - 350]j = 0i + 0j.
This gives us a system of equations:
|R|(cos125°) + |S|(cos37°) = 0,
|R|(sin125°) + |S|(sin37°) - 350 = 0.
Solving this system, we get
|R| &asymp 280 and |S| &asymp 201.
The tensions in the cables are 280 lb and 201 lb.


Correlation: Definition and Importance of Proper Data Interpretation

- Guide Authored by Corin B. Arenas, published on September 25, 2019

Ever thought of how our needs impact prices? How about your stress levels in relation to your financial habits? All these are situations that require correlation analysis.

Read on to learn more about correlation, why it&rsquos important, and how it can help you understand random connections better.

What is Correlation?

The study of how variables are related is called correlation analysis.

Correlation measures the strength of how two things are related. Britannica defines it as the degree of association between 2 random variables.

In statistics, correlational analysis is a method used to evaluate the strength of a relationship between two numerically measured, continuous variables. Unlike controlled experiments, the defining aspect of correlational studies is that neither of the variables are manipulated.

In finance, the correlation can measure the movement of a stock with that of a benchmark index.

Correlation is commonly used to test associations between quantitative variables or categorical variables . The correlation between graphs of 2 data sets signify the degree to which they are similar to each other.

  1. Quantitative variables – Refers to numeric data in statistics. Examples include percentage, decimals, map coordinates, rates, prices, etc.
  2. Categorical variables – Refers to qualitative data which are descriptions of groups or things. These are not numerical. Examples include voting preference, race, cities, hair color, favorite movie, etc.

Measuring the Strength Between 2 Variables

A correlation coefficient formula is used to determine the relationship strength between 2 continuous variables.

The formula was developed by British statistician Karl Pearson in the 1890s, which is why the value is called the Pearson correlation coefficient (r) . The equation was derived from an idea proposed by statistician and sociologist Sir Francis Galton. See the formula below:

Pearson&rsquos correlation coefficient is also known as the &lsquoproduct moment correlation coefficient&rsquo (PMCC). It has a value between -1 and 1 where:

  • A zero result signifies no relationship at all
  • 1 signifies a strong positive relationship
  • -1 signifies a strong negative relationship

What these results indicate:

  • Zero result – It means the two variables do not have any linear relation at all. Some connection may exist between the two, but not in a linear manner.
  • Positive correlation – A variable rises simultaneously with the other and moves in the same direction. High numerical figures on one set relates to high numerical figures of the other set.
  • Negative correlation – A variable decreases as the other variable increases. They move in opposite directions. High numerical figures on one set relates to the low numerical figures of the other set.

When plotted in a graph, here&rsquos how variable relationships translate visually:

Positive and Negative Numerical Relationships

When we study market trends, positive correlation is commonly found between product demand and price .

Prices increase when firms cannot produce enough supplies for the consumer&rsquos needs. This is the fundamental concept behind the law of supply and demand. Consumer spending and gross domestic product (GDP) are two variables that maintain a positive correlation with each other.

When it comes to investments, there is a positive correlation between the amount of risk and potential for return. However, there is no guarantee that taking a higher risk will often yield greater return.

To counteract this, investments with varying levels of risk are placed together in a portfolio to diversify it. This helps maximize returns while lessening the potential for large drawdowns as volatility spikes within a particular asset class.

Here are other examples of positive correlation:

  • Weight and height
  • Caloric intake and weight
  • Computer use and grade point average (GPA)
  • Child&rsquos eye color and relatives&rsquo eye color
  • Time of investment and compounding interests

In finance, a negative correlation or an inverse relationship occurs between investment returns of 2 different assets. A good example is negative correlation between equities and bonds . It indicates that bonds perform well when equities sell off.

However, note that the correlation between these variables is not static. Since it&rsquos continuous, it means the correlation may shift over time, from negative to positive, and vice versa. But for majority of the time, U.S. equities and bonds have had a negative correlation since the late 1990s.

Other examples of negative correlation include:

  • Amount of money earned and time spent with family
  • Number of cigarettes per day and lifespan
  • Cold temperatures and electricity cost (in a tropical area)
  • Amount of snow fall and number of cars on the road
  • Positive behavior in healthcare professionals and patient mortality rates
  • Positive financial habits and level of stress

Correlation vs. Causation

Correlational research models do not always indicate causal relationships.

Knowing that two variables are associated does not automatically mean one causes the other. A correlational link between two variables may simply report that their trend moves in a synchronized manner.

For a causal relationship to occur, a variable must directly cause the other.

For instance, we might establish there is a correlation between the number of roads built in the U.S. and the number of children born in the U.S. While we might see more roads being constructed and more children are being born, it does not mean the relationship is a causal one.

It leads us to consider a third hidden variable which directly affects the behavior of the two variables. If a researcher is unaware of this confounding variable, they may interpret the data incorrectly.

For this example, people might think the construction of roads causes the birth of more children. It&rsquos a ridiculous assumption, one that&rsquos often made fun of at the Spurious Correlations site.

If we think about it, the third variable causing more road constructions and child births can be attributed to the general improvement of the U.S. economy.

Flawed Research Models and Correlational Interpretations

A 2015 article in the American Scientist pointed out how misinterpretation of correlations can render research papers inaccurate and useless. It can also be dangerously misleading to medical practitioners and the public.

The story referred to a 2012 study published in the New England Journal of Medicine, claiming that chocolate consumption could boost cognitive function. Again, the correlation did not account for the nature of the quantitative link. It only presented strong similarities between the variables.

If peer reviewed journals overlook flaws in research methods and interpretation, what more with general biomedical news? The incident alarmed medical and scientific communities, calling for proper research parameters to prevent the spread of misleading information.

However, even when experts criticized the study, many news outlets still reported its findings. The paper was never retracted and has been cited several times.

It calls to mind how George E.P. Box described statistical models as oversimplifications of reality:

&ldquoEssentially, all [statistical] models are wrong, but some are useful.&rdquo

-George E. P. Box, &lsquoEmpirical Model Building and Response Surfaces&rsquo

The Takeaway

Knowing the right way to use correlations can help pinpoint what connects two variables. This in turn helps predict future trends based on the patterns they create.

However, careless use of correlation can be misleading to the public. Which is why it&rsquos important to set proper research models before using correlations to justify a study.

Correlation analysis is crucial for all sorts of fields, such as government and health care sectors. Companies also use correlations to analyze budgets and create effective business plans.