Problem 40 - Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\mathbf{U}+\mathbf{V} = \langle 0, 4 \rangle\), \(\mathbf{U}-\mathbf{V} = \langle -6, 6 \rangle\), \(2 \mathbf{U} - 3 \mathbf{V} = \langle -15, 13 \rangle\)."

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

To find \(\mathbf{U} + \mathbf{V}\), add the corresponding components of the vectors \(\mathbf{U}\) and \(\mathbf{V}\): \[ \mathbf{U} + \mathbf{V} = \langle -3 + 3, 5 + (-1) \rangle = \langle 0, 4 \rangle \].

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

Subtract the corresponding components of the vector \(\mathbf{V}\) from \(\mathbf{U}\): \[ \mathbf{U} - \mathbf{V} = \langle -3 - 3, 5 - (-1) \rangle = \langle -6, 6 \rangle \].

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

First, calculate \(2\mathbf{U}\) and \(3\mathbf{V}\), and then find their difference:1. \(2\mathbf{U} = 2 \times \langle -3, 5 \rangle = \langle -6, 10 \rangle\)2. \(3\mathbf{V} = 3 \times \langle 3, -1 \rangle = \langle 9, -3 \rangle\)3. Now subtract: \(2\mathbf{U} - 3\mathbf{V} = \langle -6, 10 \rangle - \langle 9, -3 \rangle = \langle -6 - 9, 10 - (-3) \rangle = \langle -15, 13 \rangle\).

Key Concepts

Vector Addition

In vector arithmetic, vector addition is one of the most fundamental operations. Think of it as simply combining two vectors to determine the overall result. Imagine each vector as representing a direction and magnitude—a bit like an arrow pointing somewhere. When you add these two vectors together, you're essentially placing the second vector's tail at the first vector's head and then drawing a third vector from the tail of the first vector to the head of the second. This new vector represents their sum.

To perform vector addition with components, you can follow these easy steps:

• Take each corresponding component, or part, of the vectors.
• Add them together separately for each dimension. This means adding the x-components together, the y-components together, and so on.

In our case with vectors \(\mathbf{U} = \langle -3, 5 \rangle\) and \(\mathbf{V} = \langle 3, -1 \rangle\):

• First, add the x-components: \(-3 + 3 = 0\).
• Next, add the y-components: \(5 + (-1) = 4\).

The resulting vector from adding \(\mathbf{U}\) and \(\mathbf{V}\) is \(\langle 0, 4 \rangle\). Each component is added separately and forms the new vector, illustrating how both vectors combined would move certain units along the x and y-axis.

Vector Subtraction

Vector subtraction involves removing the influence of one vector from another. You can consider this as reversing the direction of the vector being subtracted and then adding the two vectors. Essentially, subtraction is just the addition of a negative vector.

To subtract vectors using their components:

• Subtract the x-component of the second vector from the x-component of the first vector.
• Similarly, subtract the y-component of the second vector from the y-component of the first vector.

Referring to our vectors \(\mathbf{U} = \langle -3, 5 \rangle\) and \(\mathbf{V} = \langle 3, -1 \rangle\):

• Subtract the x-components: \(-3 - 3 = -6\).
• Subtract the y-components: \(5 - (-1) = 6\).

The result of subtracting \(\mathbf{V}\) from \(\mathbf{U}\) is \(\langle -6, 6 \rangle\). This gives us a vector indicating how to move from the tip of \(\mathbf{V}\) back to the base of \(\mathbf{U}\), essentially reversing part of their directions and combinations.

Scalar Multiplication

Scalar multiplication is another important operation in vector arithmetic, involving the multiplication of a vector by a scalar (a real number). This operation changes the magnitude of the vector without affecting its direction, except if the scalar is negative, which will reverse the direction.

Performing scalar multiplication involves a straightforward method:

• Multiply each component of the vector by the scalar.

Let's illustrate this with our vectors again. Consider \(2\mathbf{U}\) and \(3\mathbf{V}\):

• For \(2\mathbf{U}\): Multiply each component of \(\mathbf{U} = \langle -3, 5 \rangle\) by 2, and you get \(\langle -6, 10 \rangle\).
• For \(3\mathbf{V}\): Multiply each component of \(\mathbf{V} = \langle 3, -1 \rangle\) by 3, resulting in \(\langle 9, -3 \rangle\).

Finally, to solve \(2\mathbf{U} - 3\mathbf{V}\), subtract the resulting vectors:

• Subtract the x-components: \(-6 - 9 = -15\).
• Subtract the y-components: \(10 - (-3) = 13\).

The result is the vector \(\langle -15, 13 \rangle\), which shows the combined result of scaling and then subtracting \(\mathbf{V}\) from the double of \(\mathbf{U}\). This operation often helps in scaling movements or changes expressed by vectors.