Problem 23 - Contemporary Precalculus

Question

Determine whether the given vectors are parallel, orthogonal, or neither. $$2 \mathbf{i}-2 \mathbf{j}, 5 \mathbf{i}+8 \mathbf{j}$$

Step-by-Step Solution

Answer

Answer: Neither

Step 1: Check for parallelism

In order to determine whether two vectors, $$\mathbf{u}$$ and $$\mathbf{v}$$, are parallel, we need to check if there is a scalar constant, $$k$$, such that $$\mathbf{u}=k\mathbf{v}$$. Here, we have: $$ 2\mathbf{i} - 2\mathbf{j} = k(5\mathbf{i} + 8\mathbf{j}) $$ where k is a constant scalar value. Since the coefficients of i and j do not satisfy this condition, the two given vectors are not parallel.

Step 2: Check for orthogonality

To check if two vectors are orthogonal, we need to compute the dot product of both vectors and see if the result equals to zero. The dot product of two vectors, $$\mathbf{u} = a\mathbf{i} + b\mathbf{j}$$ and $$\mathbf{v} = c\mathbf{i} + d\mathbf{j},$$ is given by: $$ \mathbf{u} \cdot \mathbf{v} = ac + bd. $$ Using the given vectors, we have: $$ (2\mathbf{i} - 2\mathbf{j}) \cdot (5\mathbf{i} + 8\mathbf{j}) = (2\cdot 5) + (-2\cdot 8) = 10 - 16 = -6 $$ Since the dot product is not equal to zero, the given vectors are not orthogonal.

Step 3: Conclusion

Since the given vectors are neither parallel nor orthogonal, we can conclude that the given vectors are neither parallel nor orthogonal.

Key Concepts

Parallel Vectors

Parallel vectors are vectors that have the same or exactly opposite direction. One of the simplest ways to determine if two vectors are parallel is by checking if one vector is a scalar multiple of the other. In mathematical terms:

• The vector \( \mathbf{u} \) is a scalar multiple of \( \mathbf{v} \) if there exists a scalar \( k \) such that \( \mathbf{u} = k \mathbf{v} \).

For example, if you have vectors \( 2\mathbf{i} - 2\mathbf{j} \) and \( 5\mathbf{i} + 8\mathbf{j} \), then to check for parallelism, you would assign the components of these vectors in the equality \( 2 = 5k \) and \( -2 = 8k \). Solving these equations will tell you if a common scalar exists.
Since no single scalar \( k \) satisfies both conditions simultaneously here, these vectors are not parallel. If they were, both ratios of the corresponding components would have to match this scalar or its negative.

Orthogonal Vectors

Orthogonality in vectors means they are perpendicular to each other. Two vectors are said to be orthogonal if their dot product is zero. The dot product can be calculated by multiplying corresponding components and summing the results. So, for vectors \( \mathbf{u} \) and \( \mathbf{v} \), the dot product \( \mathbf{u} \cdot \mathbf{v} \) is given by the formula:

• \( \mathbf{u} \cdot \mathbf{v} = a \cdot c + b \cdot d \)

Using our vectors \( 2\mathbf{i} - 2\mathbf{j} \) and \( 5\mathbf{i} + 8\mathbf{j} \), we calculate the dot product as follows:

• \( (2 \cdot 5) + (-2 \cdot 8) = 10 - 16 = -6 \)

Since the result is not zero, the vectors are not orthogonal. Orthogonal vectors have an important property: when placed tail to tail, they form a right angle, which is geometrically distinct and very useful in calculations.

Dot Product

The dot product is a fundamental operation in vector analysis. It takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation serves various purposes, including finding angles between vectors, determining orthogonality, and measuring projection lengths. The formula for the dot product of vectors \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \) is:

• \( \mathbf{u} \cdot \mathbf{v} = ac + bd \)

It's essential to execute operations involving the dot product, keeping track of each component and their signs. For example, calculating the dot product for the vectors \( 2\mathbf{i} - 2\mathbf{j} \) and \( 5\mathbf{i} + 8\mathbf{j} \) results in:

• \( 2 \times 5 + (-2) \times 8 = 10 - 16 = -6 \)

This numerical result, \(-6\), indicates a lack of orthogonality. The dot product offers insight into the relative direction of vectors, where a positive result implies vectors are oriented in a similar direction, zero denotes perpendicularity, and a negative value indicates opposite directions.