Problem 45 - Algebra and Trigonometry

Question

Find each quantity if \(\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}\) \(\|\mathbf{v}-\mathbf{w}\|\)

Step-by-Step Solution

Answer

The magnitude of \(\textbf{v} - \textbf{w}\) is \sqrt{89}.

Step 1: Subtract the vectors

Subtract the vectors \(\textbf{v} - \textbf{w}\) by subtracting their corresponding components: \[ \textbf{v} - \textbf{w} = (3 \textbf{i} - 5 \textbf{j}) - (-2 \textbf{i} + 3 \textbf{j}) = (3 - (-2)) \textbf{i} + (-5 - 3) \textbf{j} \]

Step 2: Simplify the vector

Simplify the components: \[ (3 + 2)\textbf{i} + (-5 - 3)\textbf{j} = 5 \textbf{i} - 8 \textbf{j} \]

Step 3: Calculate the magnitude

Use the formula for magnitude \[ \| \textbf{v} - \textbf{w} \| = \sqrt{ (5)^2 + (-8)^2 } = \sqrt{ 25 + 64 } = \sqrt{ 89 } \]

Key Concepts

Magnitude Calculation

Understanding how to calculate the magnitude of a vector is essential in many areas of mathematics and physics. The magnitude essentially measures the length or size of a vector. To calculate the magnitude of a vector \(\mathbf{v}\), represented by its components \(v_x\) in the x-direction and \(v_y\) in the y-direction, you can use the Pythagorean Theorem.
The formula for the magnitude \(|\mathbf{v}|\) is: \[ |\mathbf{v}| = \sqrt{ v_x^2 + v_y^2 } \]
For example, if we have a vector \(\mathbf{z} = 5\mathbf{i} - 8\mathbf{j}\), we substitute the components into the formula: \[ |\mathbf{z}| = \sqrt{ 5^2 + (-8)^2 } = \sqrt{ 25 + 64 } = \sqrt{ 89 } \] This gives us the magnitude of \(\mathbf{z}\). Recognizing the significance of the magnitude helps in understanding distances and lengths in vector fields.

Vector Components

Vectors have components that break them down into their directions along the coordinate axes. These components help in performing vector arithmetic with ease. Each vector in a 2-dimensional space can be described using two components: one for the x-axis (i-direction) and another for the y-axis (j-direction).
For instance, the vector \( \mathbf{v} = 3 \mathbf{i} - 5 \mathbf{j} \) has components 3 and -5, respectively.
When we perform operations such as addition or subtraction with another vector, we handle each component separately. This is demonstrated in the subtraction of \(\mathbf{v} - \mathbf{w}\): \[ \mathbf{v} - \mathbf{w} = (3 \mathbf{i} - 5 \mathbf{j}) - (-2 \mathbf{i} + 3 \mathbf{j}) = (3 - (-2)) \mathbf{i} + (-5 - 3) \mathbf{j} \]
Simplifying this, we get: \[ 5 \mathbf{i} - 8 \mathbf{j} \] Understanding these components is crucial, as they provide the step-by-step instructions for vector manipulation.

Vector Arithmetic

Vector arithmetic involves operations like addition, subtraction, and scalar multiplication on vectors. Working with vector arithmetic requires understanding each vector's components and how they interact.
In vector subtraction, for example, you subtract corresponding components of the vectors. Given \(\mathbf{v} = 3 \mathbf{i} - 5 \mathbf{j}\) and \(\mathbf{w} = -2 \mathbf{i} + 3 \mathbf{j}\), subtract \(\mathbf{w}\) from \(\mathbf{v}\): \[ \mathbf{v} - \mathbf{w} = (3 - (-2)) \mathbf{i} + (-5 - 3) \mathbf{j} \] This results in: \[ 5 \mathbf{i} - 8 \mathbf{j} \]
Each component's subtraction is handled independently. These operations are fundamental in fields like physics, where vectors represent quantities like force and velocity. Grasping these basics ensures a strong foundation for more complex applications of vectors.