Lesson Objectives
• Learn how to add vectors graphically
• Learn how to add vectors algebraically
• Learn how to multiply a vector by a scalar

How to Perform Vector Operations

In this lesson, we will learn how to perform operations with vectors. Let's begin with a list of vector operations:

Vector Operations

Let a, b, c, d, and k represent real numbers.

The sum of two or more vectors is also a vector known as the resultant vector. Additionally, we can say that vector addition is commutative, this means the order in which we add will not change the result. $$\langle a, b \rangle + \langle c, d \rangle=\langle a + c, b + d \rangle$$ If we have two vectors, a, and b, the sum can be shown geometrically in two ways:

Triangle Law of Vector Addition:

1. Sketch vector a
2. Sketch vector b by placing the initial point at the terminal point of vector a
3. The resultant of vectors a and b has the same initial point as vector a and the same terminal point as vector b

Parallelogram Law of Vector Addition

1. Sketch vector a
2. Sketch vector b with the same initial point of vector a
3. Complete the parallelogram
4. The resultant of vectors a and b is the diagonal of the parallelogram with the same initial point as vectors a and b
Let's look at a few examples.
Example #1: Find the resultant of vectors t and u. Show this graphically using the triangle method. $$\overrightarrow{t}=\langle 6, -8 \rangle$$ $$\overrightarrow{u}=\langle 3, 4 \rangle$$ Using the rule for adding two vectors, we can show that: $$\overrightarrow{t}+ \overrightarrow{u}=\langle 6 + 3, -8 + 4 \rangle$$ $$\overrightarrow{t}+ \overrightarrow{u}=\langle 9, -4 \rangle$$ Let's now sketch this graphically using the triangle law of vector addition. Example #2: Find the resultant of vectors t and u. Show this graphically using the parallelogram method. $$\overrightarrow{t}=\langle -1, 8 \rangle$$ $$\overrightarrow{u}=\langle 4, -3 \rangle$$ Using the rule for adding two vectors, we can show that: $$\overrightarrow{t}+ \overrightarrow{u}=\langle -1 + 4, 8 + (-3) \rangle$$ $$\overrightarrow{t}+ \overrightarrow{u}=\langle 3, 5 \rangle$$ Let's now sketch this graphically using the parallelogram law of vector addition.

Multiplying Vectors by a Scalar

Geometrically, the product of a vector v and a scalar k is the vector that is |k| times as long as v. When k is positive, kv has the same direction as v when k is negative, kv has the opposite direction of v. $$k \cdot \langle a, b \rangle=\langle ka, kb \rangle$$ Let's look at an example.
Example #3: Find and illustrate -2v. $$\overrightarrow{v}=\langle -3, 4 \rangle$$ $$-2\overrightarrow{v}=\langle -2 \cdot -3, -2 \cdot 4 \rangle$$ $$-2\overrightarrow{v}=\langle 6, -8 \rangle$$ In some cases, we may need to combine vector operations. Let's look at an example.
Example #4: Find the vector. $$\overrightarrow{v}=\langle -2, 5 \rangle$$ $$\overrightarrow{w}=\langle 3, 4 \rangle$$ Find: $$-3\overrightarrow{v}+ 2\overrightarrow{w}$$ Let's start by finding -3v: $$-3\overrightarrow{v}=-3\langle -2, 5 \rangle$$ $$-3\overrightarrow{v}=\langle 6, -15 \rangle$$ Now, let's find 2w: $$2\overrightarrow{w}=2\langle 3, 4 \rangle$$ $$2\overrightarrow{w}=\langle 6, 8 \rangle$$ Now, let's find our sum: $$-3\overrightarrow{v}+ 2\overrightarrow{w}=\langle 6 + 6, -15 + 8 \rangle$$ $$-3\overrightarrow{v}+ 2\overrightarrow{w}=\langle 12, -7 \rangle$$ Example #5: Find the component form of f - b. $$\overrightarrow{f}:$$ Magnitude: $$|f|=17$$ Direction Angle: $$θ=212°$$ $$\overrightarrow{b}:$$ Magnitude: $$|b|=22$$ Direction Angle: $$θ=10°$$ $$\overrightarrow{f}=\langle 17 \cdot \text{cos}\hspace{.1em}212°, 17 \cdot \text{sin}\hspace{.1em}212° \rangle$$ $$\overrightarrow{f}≈ \langle -14.42, -9.01 \rangle$$ $$\overrightarrow{b}=\langle 22 \cdot \text{cos}\hspace{.1em}10°, 22 \cdot \text{sin}\hspace{.1em}10° \rangle$$ $$\overrightarrow{b}≈ \langle 21.67, 3.82\rangle$$ $$\overrightarrow{f}- \overrightarrow{b}≈ \langle -14.42 - 21.67, -9.01 - 3.82 \rangle$$ $$\overrightarrow{f}- \overrightarrow{b}≈ \langle -36.09, -12.83 \rangle$$

Skills Check:

Example #1

Find the component form. $$-2\overrightarrow{a}+ 8\overrightarrow{g}$$ $$\overrightarrow{a}=\langle 10, -12 \rangle$$ $$\overrightarrow{g}=\langle 5, 2 \rangle$$

A
$$\langle 46, 40 \rangle$$
B
$$\langle -62, 60 \rangle$$
C
$$\langle -9, -8 \rangle$$
D
$$\langle 16, 22 \rangle$$
E
$$\langle 20, 40 \rangle$$

Example #2

Find the component form. $$6\overrightarrow{f}- 5\overrightarrow{g}$$ $$\overrightarrow{f}=\langle -6, -9 \rangle$$ $$\overrightarrow{g}=\langle -7, 8 \rangle$$

A
$$\langle -81, -156 \rangle$$
B
$$\langle -1, -94 \rangle$$
C
$$\langle 100, -20 \rangle$$
D
$$\langle 53, -28 \rangle$$
E
$$\langle -19, 27 \rangle$$

Example #3

Find the component form. Round to the nearest hundredth. $$-8\overrightarrow{f}+ 3\overrightarrow{v}$$ $$\left|\hspace{.15em}\overrightarrow{f}\hspace{.25em}\right|=5$$ $$θ=121°$$ $$\left|\hspace{.15em}\overrightarrow{v}\hspace{.25em}\right|=11$$ $$θ=320°$$

A
$$\langle 45.88, -55.5 \rangle$$
B
$$\langle -190.36, 43.26 \rangle$$
C
$$\langle 94.17, -45.8 \rangle$$
D
$$\langle 94.51, 91.93 \rangle$$
E
$$\langle 27.51, -12.2 \rangle$$

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