Lesson 5 The Arithmetic of Vectors Solidify Understanding

Jump Start

Notice and Wonder:

Examine the following list of quantities, and describe at least two things you notice and something you are wondering about:

A book costs $5 dollars$ .
A $30 miles per hour$ wind blowing towards the northeast.
$200 square inches$ on a bookshelf.
The force of gravity accelerates objects at $32 \frac{ft}{{sec}^{2}}$ toward the ground.
A plane flying $520 \frac{miles}{hour}$ directly west.
$12 grams$ of silver.

Learning Focus

Represent quantities that have both magnitude and direction using vectors, and examine the arithmetic of vectors.

How do we represent quantities that have both magnitude (size) and direction?

How do we add or subtract quantities that have both magnitude and direction?

Open Up the Math: Launch, Explore, Discuss

The following diagram shows a triangle that has been translated to a new location, and then translated again. Arrows have been used to indicate the movement of one of the vertex points through each translation. The result of the two translations can also be thought of as a single translation, as shown by the third arrow in the diagram.

1.

Draw arrows to show the movement of the other two vertices through the sequence of translations, and then draw an arrow to represent the resultant single translation. What do you notice about each set of arrows?

A vector is a quantity that has both magnitude and direction. The arrows you drew on the diagram represent translations as vectors—each translation has magnitude (the distance moved) and direction (the direction in which the object is moved). Arrows, or directed line segments, are one way of representing a vector.

Addition of Vectors

In the example provided, two vectors $\vec{v}$ and $\vec{w}$ were combined to form vector $\vec{r}$ . This is what is meant by “adding vectors.”

2.

Study each of the following three methods for adding vectors, then try each method to add vectors $\vec{s}$ and $\vec{t}$ given in the diagrams to find $\vec{q}$ , such that $\vec{s} + \vec{t} = \vec{q}$ .

a.

Method 1: End-to-end

The diagram given in problem 1 illustrates the end-to-end strategy of adding two vectors to get a resultant vector that represents the sum of the two vectors. In this case, the resulting vector shows that a single translation could accomplish the same movement as the combined sum of the two individual translations, that is, $\vec{v} + \vec{w} = \vec{r}$ .

b.

Method 2: The parallelogram rule

Since we can relocate the arrow representing a vector, draw both vectors starting at a common point. Often both vectors are relocated so they have their tail ends at the origin. These arrows form two sides of a parallelogram. Draw the other two sides. The resulting sum is the vector represented by the arrow drawn from the common starting point (for example, the origin) to the opposite vertex of the parallelogram.

Question to think about: How can you determine where to put the missing vertex point of the parallelogram?

c.

Method 3: Using horizontal and vertical components

Each vector consists of a horizontal component and a vertical component. For example, vector $\vec{v}$ can be thought of as a movement of $8$ units horizontally and $13$ units vertically. This is represented with the notation $⟨ 8, 13 ⟩$ . Vector $\vec{w}$ consists of a movement of $7$ units horizontally and $- 5$ units vertically, represented by the notation $⟨ 7, - 5 ⟩$ .

d.

Explain why each of these methods gives the same result.

Question to think about: How can the components of the individual vectors be combined to determine the horizontal and vertical components of the resulting vector $\vec{r}$ ?

3.

Examine vector $\vec{s}$ given. While we can relocate the vector, in the diagram the tail of the vector is located at $(3, 2)$ and the head of the vector is located at $(5, 7)$ . Explain how you can determine the horizontal and vertical components of a vector from just the coordinates of the point at the tail and the point at the head of the vector. That is, how can you find the horizontal and vertical components of movement without counting across and up the grid?

Magnitude of Vectors

The symbol $‖ \overset{⇀}{v} ‖$ is used to denote the magnitude of the vector, in this case the length of the vector.

Using the diagram given at the beginning of the task, devise a method for finding the magnitude of a vector, and use your method to find the following. Be prepared to describe your method for finding the magnitude of a vector.

4.

$‖ \vec{v} ‖$

5.

$‖ \vec{w} ‖$

6.

$‖ \vec{v} + \vec{w} ‖$

Scalar Multiples of Vectors

A vector can be stretched by multiplying the vector by a scale factor. For example, $2 \overset{⇀}{v}$ represents the vector that has the same direction as $\overset{⇀}{v}$ , but whose magnitude is twice that of $\overset{⇀}{v}$ .

Draw each of the following vectors on a coordinate grid:

7.

$3 \vec{s}$

8.

$- 2 \vec{t}$

9.

$3 \vec{s} + (- 2 \vec{t})$

10.

$3 \vec{s} - 2 \vec{t}$

Other Applications of Vectors

The concept of a vector has been illustrated using translation vectors in which the magnitude of the vector represents the distance a point gets translated. There are other quantities that have magnitude and direction, but the magnitude of the vector does not always represent length.

For example, a car traveling $55 miles per hour$ along a straight stretch of highway can be represented by a vector since the speed of the car has magnitude, $55 miles per hour$ , and the car is traveling in a specific direction. Pushing on an object with $25 pounds$ of force is another example. A vector can be used to represent this push since the force of the push has magnitude, $25 pounds$ of force, and the push would be in a specific direction.

11.

A swimmer is swimming across a river with a speed of $20 \frac{ft}{sec}$ and at a $45^{\circ}$ angle from the bank of the river. The river is flowing at a speed of $5 \frac{ft}{sec}$ . Illustrate this situation with a vector diagram, and describe the meaning of the vector that represents the sum of the two vectors that represent the motion of the swimmer and the flow of the river.

12.

Two teams are participating in a tug-of-war. One team exerts a combined force of $200 pounds$ in one direction while the other team exerts a combined force of $150 pounds$ in the other direction. Illustrate this situation with a vector diagram, and describe the meaning of the vector that represents the sum of the vectors that represent the efforts of the two teams.

Ready for More?

List all of the quantities you can think of that involve both magnitude and direction. Then create a scenario based on one of your types of vector quantities, like the scenarios in problems 11 and 12. You might want to consult with the physics teacher at your school for examples.

Takeaways

There are three strategies I can use to add vectors:

To subtract vectors:

To multiply a vector by a scalar:

Adding Notation, Vocabulary, and Conventions

Scalar quantities:

Vector quantities:

Vocabulary

Lesson Summary

In this lesson, we learned how to represent quantities that have both magnitude and direction, such as a wind blowing at $30 mph$ towards the northeast, as a directed line segment, or vector, on a coordinate grid. We also learned how to add and subtract vector quantities, and examined contexts where vector arithmetic is useful.

Retrieval

1.

Rotate point $E$ $90 °$ counterclockwise about the origin, and label it $E^{'}$ .
Rotate point $E$ $180 °$ counterclockwise about the origin, and label it $E^{″}$ .
Find the equation of the circle that passes through $E$ , $E^{'}$ , and $E^{″}$ .

2.

Multiply the matrices.

$[\begin{array}{rr} 2 & 3 \\ - 5 & 1 \\ 7 & - 4 \end{array}] \cdot [\begin{array}{rr} 1 & 0 \\ - 1 & 1 \end{array}]$