Problem 47 - Algebra and Trigonometry

Question

Determine whether \(\mathbf{v}\) and \(\mathbf{w}\) are parallel, orthogonal, or neither. $$ \mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+10 \mathbf{j} $$

Step-by-Step Solution

Answer

The given vectors \(\mathbf{v}\) = \(3\mathbf{i} - 5\mathbf{j}\) and \(\mathbf{w}\) = \(6\mathbf{i} + 10\mathbf{j}\) are neither parallel nor orthogonal.

Step 1: Check for Parallelism

Calculate the ratio of the corresponding coefficients of vectors \(\mathbf{v}\) and \(\mathbf{w}\). The two vectors are parallel if there exists a non-zero scalar such that if it is multiplied with \(\mathbf{v}\), the result is \(\mathbf{w}\). \(\frac{6}{3} = 2\) and \(\frac{10}{-5} = -2\). The ratios are not equal, so the vectors are not parallel.

Step 2: Check for Orthogonality

Calculate the dot product of \(\mathbf{v}\) and \(\mathbf{w}\). If the dot product is 0, the vectors are orthogonal. In this case, \(\mathbf{v} \cdot \mathbf{w} = (3 * 6) + (-5 * 10) = -32\). As the dot product is non-zero, the vectors are not orthogonal.

Step 3: Determine final status

Since the \(\mathbf{v}\) and \(\mathbf{w}\) are neither parallel (as the scalar multiples do not give the same ratio) nor orthogonal (as the dot product was not 0), then they are neither parallel nor orthogonal.

Key Concepts

Parallel Vectors

Vectors are known to be parallel if they point in the same or completely opposite direction. Two vectors \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{w} = c\mathbf{i} + d\mathbf{j} \) are parallel if one is a scalar multiple of the other. - This means, for instance, that \( \mathbf{w} = k \cdot \mathbf{v} \) for some non-zero scalar \( k \).- Or simply, the ratio \( \frac{c}{a} \) must be the same as \( \frac{d}{b} \).Let's break down the calculation with the given vectors:

1. Calculate \( \frac{c}{a} = \frac{6}{3} = 2 \).
2. Calculate \( \frac{d}{b} = \frac{10}{-5} = -2 \).

The ratios are not equal (one is 2, the other is -2), hence the vectors are not parallel.When checking parallelism, always assess whether both ratios are equal. If not, the vectors do not align in the same or opposite directions.

Orthogonal Vectors

Orthogonal vectors are vectors that are perpendicular to each other. This is mathematically determined by their dot product. When the dot product of two vectors is zero, the vectors are orthogonal. To compute this:1. Take two vectors, \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{w} = c\mathbf{i} + d\mathbf{j} \).2. Calculate their dot product: \( \mathbf{v} \cdot \mathbf{w} = ac + bd \).Now, using the given vectors, calculate the following:- \( \mathbf{v} \cdot \mathbf{w} = (3 \times 6) + (-5 \times 10) = 18 - 50 = -32 \).Since -32 is not zero, \( \mathbf{v} \) and \( \mathbf{w} \) are not orthogonal.In general, for vectors to be perpendicular, their interaction should result in a sum product of zero.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector analysis. It provides a way to multiply two vectors and obtain a scalar. The dot product of two vectors \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{w} = c\mathbf{i} + d\mathbf{j} \) is calculated as:\[\mathbf{v} \cdot \mathbf{w} = ac + bd\]It basically combines the effects of both magnitude and direction:- For vectors with components, multiply corresponding components together and add them up.- If the result, or the dot product, is zero, the vectors are orthogonal.In our specific scenario, the computation was:- \( (3 \times 6) + (-5 \times 10) = 18 - 50 = -32 \), which is not zero.This confirms the vectors are not orthogonal.The dot product is crucial in determining angles between vectors and can reveal if they are perpendicular.