Problem 16 - Algebra and Trigonometry

Question

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=3 \mathbf{i}-4 \mathbf{j}, \quad \mathbf{w}=9 \mathbf{i}-12 \mathbf{j} $$

Step-by-Step Solution

Answer

The dot product is 75. The angle between the vectors is 0 degrees. The vectors are parallel.

Step 1 - Write down the given vectors

Given vectors are oindent oindent \( \mathbf{v} = 3 \mathbf{i} - 4 \mathbf{j} \) and \( \mathbf{w} = 9 \mathbf{i} - 12 \mathbf{j} \)

Step 2 - Find the dot product of \( \mathbf{v} \) and \( \mathbf{w} \)

Calculate using the dot product formula: \( \mathbf{v} \cdot \mathbf{w} = 3 \times 9 + (-4) \times (-12) \)Simplify the calculation: \( \mathbf{v} \cdot \mathbf{w} = 27 + 48 = 75 \)

Step 3 - Calculate magnitudes of \( \mathbf{v} \) and \( \mathbf{w} \)

Find the magnitude of each vector: \( \| \mathbf{v} \| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \) \( \| \mathbf{w} \| = \sqrt{9^2 + (-12)^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \)

Step 4 - Find the cosine of the angle between \( \mathbf{v} \) and \( \mathbf{w} \)

Use the cosine formula for the angle between two vectors: \( \cos \theta = \frac{ \mathbf{v} \cdot \mathbf{w} }{ \| \mathbf{v} \| \| \mathbf{w} \| } \)Substitute the values: \( \cos \theta = \frac{75}{5 \times 15} = \frac{75}{75} = 1 \) \( \theta = \cos^{-1}(1) = 0 \, \text{degrees} \)

Step 5 - Determine if vectors are parallel, orthogonal, or neither

Since \( \cos \theta = 1 \) and \( \theta = 0 \) degrees, the vectors \( \mathbf{v} \) and \( \mathbf{w} \) are parallel.

Key Concepts

vector algebra

Vectors are fundamental in mathematics and physics to represent quantities having both magnitude and direction. In vector algebra, vectors like \( \mathbf{v} = 3 \mathbf{i} - 4 \mathbf{j} \) and \( \mathbf{w} = 9 \mathbf{i} - 12 \mathbf{j} \) can be manipulated through various operations, such as addition, subtraction, and scaling. A critical operation within vector algebra is the dot product.

angle between vectors

The angle between two vectors \( \mathbf{v} \) and \( \mathbf{w} \) can be found using their dot product. The cosine of this angle can be calculated with the formula: \[ \cos\(\theta\) = \frac{ \mathbf{v} \cdot \mathbf{w} }{ \|\mathbf{v}\| \|\mathbf{w}\| } \] In our exercise, the formula was used to find that \( \cos \theta = 1 \) and therefore \( \theta = \cos^{-1}(1) = 0 \) degrees.

parallel vs orthogonal vectors

To determine the relationship between vectors, consider the results from the dot product and the angle between them. If the angle \( \theta = 0 \) or \( \theta = 180 \) degrees, the vectors are parallel, meaning they lie along the same or exactly opposite directions. If the dot product is zero, the vectors are orthogonal.