Problem 46 - Algebra and Trigonometry

Question

Determine whether \(\mathbf{v}\) and \(\mathbf{w}\) are parallel, orthogonal, or neither. $$ \mathbf{v}=-2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}+9 \mathbf{j} $$

Step-by-Step Solution

Answer

The vectors \(\mathbf{v}\) and \(\mathbf{w}\) are parallel.

Step 1: Calculate the dot product

Calculate the dot product (\(\mathbf{v} \cdot \mathbf{w}\)) of the two vectors. This can be done by multiplying their corresponding i and j components, and then adding the results: \((-2)(-6) + (3)(9)\)

Step 2: Analyze the dot product

If the dot product is zero, the vectors are orthogonal. If the dot product is not zero, proceed to Step 3.

Step 3: Establish Ratio

Establish a ratio between the corresponding components of the two vectors, i.e., determine if \(\frac{-2}{-6} = \frac{3}{9}\). If they're equal, then \(\mathbf{v}\) and \(\mathbf{w}\) are parallel.

Step 4: Determine Relationship

If neither the dot product is zero nor the ratios are equal, the vectors are neither parallel nor orthogonal.

Key Concepts

Dot Product

The dot product is an essential tool in vector analysis for understanding relationships between vectors. It is calculated by multiplying the corresponding components of two vectors and summing these products. For the vectors \( \mathbf{v} = -2 \mathbf{i} + 3 \mathbf{j} \) and \( \mathbf{w} = -6 \mathbf{i} + 9 \mathbf{j} \), the dot product is calculated as follows:
\[ \mathbf{v} \cdot \mathbf{w} = (-2)(-6) + (3)(9) \]
This results in \( 12 + 27 = 39 \).

• If the dot product is zero, the vectors are orthogonal, indicating they meet at a right angle.
• If the dot product is not zero, as in this case, it suggests another type of relationship which can be explored further.

Parallel Vectors

Parallel vectors share the same or exact opposite direction. To determine if two vectors are parallel, we can compare the ratios of their corresponding components. If these ratios are equal, the vectors are parallel. For \( \mathbf{v} = -2 \mathbf{i} + 3 \mathbf{j} \) and \( \mathbf{w} = -6 \mathbf{i} + 9 \mathbf{j} \), we consider the ratio:
\[ \frac{-2}{-6} = \frac{3}{9} \]
This simplifies to \( \frac{1}{3} \) in both cases, indicating that \( \mathbf{v} \) and \( \mathbf{w} \) are parallel.
Here are a few points to remember:

• Parallel vectors have proportional components.
• If vectors are parallel, their direction may be the same, or they may be antiparallel (facing opposite directions).

Orthogonal Vectors

Orthogonal vectors are vectors that are perpendicular to each other, forming a right angle. This mutual perpendicularity is characterized by a dot product of zero. For vectors \( \mathbf{v} \) and \( \mathbf{w} \), we calculated their dot product:
\[ \mathbf{v} \cdot \mathbf{w} = 39 \]
Since 39 is not zero, the vectors \( \mathbf{v} \) and \( \mathbf{w} \) are not orthogonal. This conclusion tells us:

• Orthogonal vectors reveal an angle of 90 degrees between them.
• They are neither closely related directionally nor proportional in terms of component ratio.

Component Ratio

Component ratios help determine the directional relationship between vectors. This technique involves examining ratios of vectors' individual components. For vectors \( \mathbf{v} = -2 \mathbf{i} + 3 \mathbf{j} \) and \( \mathbf{w} = -6 \mathbf{i} + 9 \mathbf{j} \), we use:
\[ \frac{-2}{-6} = \frac{3}{9} \]
This simplifies to \( \frac{1}{3} \) across both comparisons, indicating that the vectors \( \mathbf{v} \) and \( \mathbf{w} \) are parallel. Key takeaways include:

• Matching component ratios signal parallel vectors.
• When not equal, the vectors are neither parallel nor essentially directionally related.