Problem 36 - Trigonometry

Question

For each vector, find \(\frac{1}{2} \mathbf{V},-\mathbf{V}\), and \(4 \mathbf{V}\). $$V=\langle-2,5\rangle$$

Step-by-Step Solution

Answer

\(\frac{1}{2} \mathbf{V} = \langle -1, 2.5 \rangle\), \(-\mathbf{V} = \langle 2, -5 \rangle\), \(4\mathbf{V} = \langle -8, 20 \rangle\).

Step 1: Calculate \(\frac{1}{2} \mathbf{V}\)

To find \(\frac{1}{2} \mathbf{V}\), multiply each component of the vector \(\mathbf{V} = \langle -2, 5 \rangle\) by \(\frac{1}{2}\). This results in: \(\frac{1}{2} \langle -2, 5 \rangle = \langle \frac{1}{2} \times -2, \frac{1}{2} \times 5 \rangle = \langle -1, 2.5 \rangle\)

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

To find \(-\mathbf{V}\), multiply each component of \(\mathbf{V} = \langle -2, 5 \rangle\) by \(-1\). This results in: \(-\langle -2, 5 \rangle = \langle -1 \times -2, -1 \times 5 \rangle = \langle 2, -5 \rangle\)

Step 3: Calculate \(4\mathbf{V}\)

To find \(4\mathbf{V}\), multiply each component of \(\mathbf{V} = \langle -2, 5 \rangle\) by \(4\). This results in: \(4 \langle -2, 5 \rangle = \langle 4 \times -2, 4 \times 5 \rangle = \langle -8, 20 \rangle\)

Key Concepts

Scalar Multiplication

Scalar multiplication is fundamental in vector mathematics and is quite straightforward. You multiply a vector by a scalar (a single number), affecting the vector's magnitude but not its direction. For example, if you have a vector \( \mathbf{V} = \langle -2, 5 \rangle \):

• To calculate \( \frac{1}{2} \mathbf{V} \), each component of the vector is multiplied by \( \frac{1}{2} \). Performing this, you get: \( \langle \frac{1}{2} \times -2 , \frac{1}{2} \times 5 \rangle = \langle -1, 2.5 \rangle \).
• Similarly, to find \( 4 \mathbf{V} \), multiply each component by \( 4 \): \( \langle 4 \times -2, 4 \times 5 \rangle = \langle -8, 20 \rangle \).

It's important to note that multiplying by a positive scalar changes the vector length proportionally, while the direction stays the same. Multiplying by a negative scalar, on the other hand, reverses the vector's direction.

Vector Components

Understanding the components of a vector helps you work efficiently with vectors in different operations. A vector in two-dimensional space like \( \mathbf{V} = \langle -2, 5 \rangle \) consists of two components: a horizontal and a vertical one. These can also be described as the x-component and the y-component:

• The x-component is \( -2 \), indicating the horizontal movement.
• The y-component is \( 5 \), showing the vertical movement.

Breaking down a vector into its components can help you visualize the direction and magnitude in the plane. This is especially useful in problems where you're asked to apply operations like scaling or finding negative vectors, as you manipulate each part separately.

Negative Vectors

Understanding negative vectors involves realizing how direction changes but the magnitude remains the same. When you create a negative vector by multiplying each component by \(-1\), you essentially flip the direction of the vector. For example:

• For \( \mathbf{V} = \langle -2, 5 \rangle \), the negative \(-\mathbf{V}\) is calculated as: \( \langle -1 \times -2, -1 \times 5 \rangle = \langle 2, -5 \rangle \).

Here, both the x-component and the y-component change their signs, reversing the direction on the coordinate plane. However, the magnitude remains the same, and it still reflects how far the vector reaches. Negative vectors are crucial for understanding vector subtraction and addition, providing flexibility in navigating space.