Problem 43 - Trigonometry

Question

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(2 \mathbf{U}-3 \mathbf{V}\). $$\mathbf{U}=\langle 4,1\rangle, \mathbf{V}=\langle-5,2\rangle$$

Step-by-Step Solution

Answer

\( \mathbf{U} + \mathbf{V} = \langle -1, 3 \rangle \), \( \mathbf{U} - \mathbf{V} = \langle 9, -1 \rangle \), \( 2\mathbf{U} - 3\mathbf{V} = \langle 23, -4 \rangle \).

Step 1: Calculate \( \mathbf{U} + \mathbf{V} \)

To find \( \mathbf{U} + \mathbf{V} \), add the corresponding components of \( \mathbf{U} = \langle 4, 1 \rangle \) and \( \mathbf{V} = \langle -5, 2 \rangle \).\\( \mathbf{U} + \mathbf{V} = \langle 4 + (-5), 1 + 2 \rangle = \langle -1, 3 \rangle \).

Step 2: Calculate \( \mathbf{U} - \mathbf{V} \)

To find \( \mathbf{U} - \mathbf{V} \), subtract the corresponding components of \( \mathbf{V} = \langle -5, 2 \rangle \) from \( \mathbf{U} = \langle 4, 1 \rangle \).\\( \mathbf{U} - \mathbf{V} = \langle 4 - (-5), 1 - 2 \rangle = \langle 9, -1 \rangle \).

Step 3: Calculate \( 2 \mathbf{U} - 3 \mathbf{V} \)

First, calculate \( 2 \mathbf{U} \) by multiplying each component of \( \mathbf{U} = \langle 4, 1 \rangle \) by 2: \( 2 \mathbf{U} = \langle 8, 2 \rangle \).\Next, calculate \( 3 \mathbf{V} \) by multiplying each component of \( \mathbf{V} = \langle -5, 2 \rangle \) by 3: \( 3 \mathbf{V} = \langle -15, 6 \rangle \).\Then, find \( 2 \mathbf{U} - 3 \mathbf{V} \) by subtracting \( 3 \mathbf{V} \) from \( 2 \mathbf{U} \).\\( 2 \mathbf{U} - 3 \mathbf{V} = \langle 8 - (-15), 2 - 6 \rangle = \langle 23, -4 \rangle \).

Key Concepts

Component-wise Operations

When dealing with vectors, one of the simplest operations we can perform is adding or subtracting them through component-wise operations. This operation involves performing arithmetic on each corresponding component of the vectors.For example, given two vectors \( \mathbf{U} = \langle 4, 1 \rangle \) and \( \mathbf{V} = \langle -5, 2 \rangle \), to find the sum \( \mathbf{U} + \mathbf{V} \), you simply add the first components together and then the second components together:

• For the first component: \( 4 + (-5) = -1 \)
• For the second component: \( 1 + 2 = 3 \)

Thus, the result is \( \mathbf{U} + \mathbf{V} = \langle -1, 3 \rangle \).
Similarly, if you wish to calculate \( \mathbf{U} - \mathbf{V} \), you subtract the second vector's components from the first vector's components:

• For the first component: \( 4 - (-5) = 9 \)
• For the second component: \( 1 - 2 = -1 \)

The result is \( \mathbf{U} - \mathbf{V} = \langle 9, -1 \rangle \).
Remember that component-wise operations follow straightforward arithmetic rules, making them valuable tools for manipulating vectors in physics and engineering.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar, which is simply a single number. This operation is crucial in stretching or shrinking a vector while maintaining its direction.To see how this works, let's examine how to compute \( 2 \mathbf{U} \) when \( \mathbf{U} = \langle 4, 1 \rangle \). Multiply each component of \( \mathbf{U} \) by 2:

• The first component becomes \( 4 \times 2 = 8 \)
• The second component becomes \( 1 \times 2 = 2 \)

Thus, \( 2 \mathbf{U} = \langle 8, 2 \rangle \).
Similarly, if asked to perform a similar operation on vector \( \mathbf{V} = \langle -5, 2 \rangle \) by multiplying by 3, we calculate:

• The first component becomes \( -5 \times 3 = -15 \)
• The second component becomes \( 2 \times 3 = 6 \)

This yields \( 3 \mathbf{V} = \langle -15, 6 \rangle \).
Scalar multiplication is especially useful when combining operations, such as in physics where vectors may need to account for varying quantities!

Vector Algebra

Vector algebra is a broad field, but at its core, it involves operations like addition, subtraction, and multiplication with scalars to manage vectors. By combining these operations, we can solve complex problems.Let's take a closer examination of \( 2 \mathbf{U} - 3 \mathbf{V} \). This involves using both scalar multiplication and component-wise operations.
The steps are as follows:

• First, compute \( 2 \mathbf{U} = \langle 8, 2 \rangle \)
• Next, compute \( 3 \mathbf{V} = \langle -15, 6 \rangle \)
• Finally, subtract these resulting vectors: \( \langle 8, 2 \rangle - \langle -15, 6 \rangle \)

By subtracting component-wise:

• The first component: \( 8 - (-15) = 23 \)
• The second component: \( 2 - 6 = -4 \)

Hence, \( 2 \mathbf{U} - 3 \mathbf{V} = \langle 23, -4 \rangle \).
Vector algebra provides a framework to manipulate and use vectors efficiently in everything from theoretical math to practical applications in engineering and navigation.