Operations with Vectors Lesson

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.

Adding Two Vectors

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: $$θ=10°$$ $$\overrightarrow{f}=\langle 17 \cdot \text{cos}\hspace{.1em}212°, 17 \cdot \text{sin}\hspace{.1em}212° \rangle$$ $$\overrightarrow{f}≈ \langle -14.42, -9.01 \rangle$$ $$\overrightarrow{b}=\langle 22 \cdot \text{cos}\hspace{.1em}10°, 22 \cdot \text{sin}\hspace{.1em}10° \rangle$$ $$\overrightarrow{b}≈ \langle 21.67, 3.82\rangle$$ $$\overrightarrow{f}- \overrightarrow{b}≈ \langle -14.42 - 21.67, -9.01 - 3.82 \rangle$$ $$\overrightarrow{f}- \overrightarrow{b}≈ \langle -36.09, -12.83 \rangle$$

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Skills Check: