Problem 24 - Thomas Calculus

Question

Parallel and perpendicular vectors Let \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}-\mathbf{k},\) \(\mathbf{v}=-\mathbf{i}+\mathbf{j}+\mathbf{k}, \mathbf{w}=\mathbf{i}+\mathbf{k}, \mathbf{r}=-(\pi / 2) \mathbf{i}-\pi \mathbf{j}+(\pi / 2) \mathbf{k}\) . Which vectors, if any, are (a) perpendicular? (b) Parallel? Give reasons for your answers.

Step-by-Step Solution

Answer

Vectors \( \mathbf{u} \) & \( \mathbf{v} \), \( \mathbf{u} \) & \( \mathbf{w} \), \( \mathbf{v} \) & \( \mathbf{w} \), \( \mathbf{v} \) & \( \mathbf{r} \), \( \mathbf{w} \) & \( \mathbf{r} \) are perpendicular; none are parallel.

Step 1: Understanding Dot Product for Perpendicularity

To check if two vectors are perpendicular, we use the dot product. If the dot product of two vectors is zero, then the vectors are perpendicular.

Step 2: Calculating Dot Products

Compute the dot product for each pair of vectors: \( \mathbf{u} \cdot \mathbf{v} \), \( \mathbf{u} \cdot \mathbf{w} \), \( \mathbf{u} \cdot \mathbf{r} \), \( \mathbf{v} \cdot \mathbf{w} \), \( \mathbf{v} \cdot \mathbf{r} \), and \( \mathbf{w} \cdot \mathbf{r} \).

Step 3: Dot Product of \( \mathbf{u} \) and \( \mathbf{v} \)

Calculate: \[ \mathbf{u} \cdot \mathbf{v} = (1)(-1) + (2)(1) + (-1)(1) = -1 + 2 - 1 = 0 \]The vectors \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular because the dot product is zero.

Step 4: Dot Product of \( \mathbf{u} \) and \( \mathbf{w} \)

Calculate: \[ \mathbf{u} \cdot \mathbf{w} = (1)(1) + (2)(0) + (-1)(1) = 1 + 0 - 1 = 0 \]The vectors \( \mathbf{u} \) and \( \mathbf{w} \) are perpendicular because the dot product is zero.

Step 5: Dot Product of \( \mathbf{u} \) and \( \mathbf{r} \)

Calculate: \[ \mathbf{u} \cdot \mathbf{r} = (1)(-\frac{\pi}{2}) + (2)(-\pi) + (-1)(\frac{\pi}{2}) = -\frac{\pi}{2} - 2\pi - \frac{\pi}{2} = -3\pi \]The vectors \( \mathbf{u} \) and \( \mathbf{r} \) are not perpendicular because the dot product is not zero.

Step 6: Dot Product of \( \mathbf{v} \) and \( \mathbf{w} \)

Calculate: \[ \mathbf{v} \cdot \mathbf{w} = (-1)(1) + (1)(0) + (1)(1) = -1 + 0 + 1 = 0 \]The vectors \( \mathbf{v} \) and \( \mathbf{w} \) are perpendicular because the dot product is zero.

Step 7: Dot Product of \( \mathbf{v} \) and \( \mathbf{r} \)

Calculate: \[ \mathbf{v} \cdot \mathbf{r} = (-1)(-\frac{\pi}{2}) + (1)(-\pi) + (1)(\frac{\pi}{2}) = \frac{\pi}{2} - \pi + \frac{\pi}{2} = 0 \]The vectors \( \mathbf{v} \) and \( \mathbf{r} \) are perpendicular because the dot product is zero.

Step 8: Dot Product of \( \mathbf{w} \) and \( \mathbf{r} \)

Calculate: \[ \mathbf{w} \cdot \mathbf{r} = (1)(-\frac{\pi}{2}) + (0)(-\pi) + (1)(\frac{\pi}{2}) = -\frac{\pi}{2} + 0 + \frac{\pi}{2} = 0 \]The vectors \( \mathbf{w} \) and \( \mathbf{r} \) are perpendicular because the dot product is zero.

Step 9: Checking for Parallel Vectors using Scalar Multiples

Two vectors are parallel if one is a scalar multiple of the other. Compare each vector to check if any is a scalar multiple of another.

Step 10: Parallel Vector Analysis

Upon comparing, none of the vectors are scalar multiples of the others, hence none are parallel.

Key Concepts

Perpendicular Vectors

Perpendicular vectors are fascinating because they meet at right angles, much like the corner of a square or rectangle. In vector analysis, two vectors are perpendicular if their dot product equals zero.
The dot product is calculated by multiplying corresponding components of two vectors and summing the results. For example, if you have vectors \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} - \mathbf{k} \) and \( \mathbf{v} = -\mathbf{i} + \mathbf{j} + \mathbf{k} \), their dot product is \( \mathbf{u} \cdot \mathbf{v} = (1)(-1) + (2)(1) + (-1)(1) = 0 \). Since the result is zero, these vectors are perpendicular.
This concept is not just about satisfying a mathematical condition, but also about understanding how vectors relate to each other in space, providing insight into geometry and physics.

Parallel Vectors

Parallel vectors are like a pair of trails that never intersect, always maintaining the same direction. Two vectors are parallel when one is a scalar multiple of the other. This means that even if one vector is longer or shorter, if you can multiply the components of one vector by the same number to get the other vector, they are parallel.
For instance, if vector \( \mathbf{a} \) is twice as long as vector \( \mathbf{b} \), they point in the same direction and we say \( \mathbf{a} \) is parallel to \( \mathbf{b} \).
In our exercise, it was examined whether any of the vectors given could be scalar multiples, and thus parallel. The analysis showed that none of the vectors fit this condition, confirming that none were parallel. Understanding parallel vectors is crucial in navigation, physics, and various fields of engineering where direction is key.

Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors to get a scalar, or a single number. This operation is foundational in vector analysis, providing essential information about the relationship between vectors.
To calculate the dot product of vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \), you multiply and sum their respective components:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \].
This operation is highly valuable as it reveals if vectors are perpendicular and can also be used to find the angle between vectors. When the dot product is zero, it means the vectors are orthogonal (perpendicular), which is an important relationship in both geometry and physics.

Scalar Multiplication

Scalar multiplication involves taking a vector and multiplying each of its components by a scalar, a real number. This operation scales the vector without changing its direction if the scalar is positive. If the scalar is negative, it also reverses the direction.
For example, multiplying the vector \( \mathbf{a} = \mathbf{i} + 2\mathbf{j} \) by a scalar \( c = 3 \) results in \( 3\mathbf{a} = 3\mathbf{i} + 6\mathbf{j} \). This new vector points in the same direction but is three times longer.
Scalar multiplication plays a role in determining parallel vectors, as any pair of vectors where one is a scalar multiple of the other is parallel. It's a fundamental concept that aids in transforming vector lengths and in practical applications like physics, where it can represent changes in physical quantities such as force or velocity.