Problem 2 - Active Calculus Multivariable

Question

Determine if the pairs of vectors below are "parallel", "orthogonal", or "neither". $$ \begin{array}{l} \mathbf{a}=\langle-1,-2,2\rangle \text { and } \mathbf{b}=\langle 4,8,10\rangle \text { are } \\ \mathbf{a}=\langle-1,-2,2\rangle \text { and } \mathbf{b}=\langle 4,8,-8\rangle \text { are } \\ \mathbf{a}=\langle-1,-2,2\rangle \text { and } \mathbf{b}=\langle 2,4,-5\rangle \text { are } \end{array} $$

Step-by-Step Solution

Answer

The pairs of vectors are classified as follows: 1. Orthogonal 2. Neither 3. Neither

Step 1: Pair 1: \(\mathbf{a}=\langle -1,-2,2\rangle \) and \(\mathbf{b}=\langle 4,8,10\rangle \)

Step 1. Check for parallel vectors Verify if vector \(\mathbf{a}\) is a multiple of vector \(\mathbf{b}\) or vice versa: \[ \begin{array}{l} \mathbf{a}=\alpha \mathbf{b} \\ \langle -1,-2,2\rangle = \alpha \langle 4,8,10\rangle \end{array} \] We can see that \(\mathbf{a}\) is not a multiple of \(\mathbf{b}\), so they are not parallel. Step 2. Check for orthogonal vectors Calculate the dot product of \(\mathbf{a}\) and \(\mathbf{b}\): \( \mathbf{a} \cdot \mathbf{b} = -1(4) + (-2)(8) + 2(10) = -4 -16 + 20 = 0 \) Since the dot product is equal to zero, the vectors are orthogonal. Pair 1: Orthogonal

Step 2: Pair 2: \(\mathbf{a}=\langle -1,-2,2\rangle \) and \(\mathbf{b}=\langle 4,8,-8\rangle \)

Step 1. Check for parallel vectors Verify if vector \(\mathbf{a}\) is a multiple of vector \(\mathbf{b}\) or vice versa: \[ \begin{array}{l} \mathbf{a}=\alpha \mathbf{b} \\ \langle -1,-2,2\rangle = \alpha \langle 4,8,-8\rangle \end{array} \] We can see that \(\mathbf{a}\) is not a multiple of \(\mathbf{b}\), so they are not parallel. Step 2. Check for orthogonal vectors Calculate the dot product of \(\mathbf{a}\) and \(\mathbf{b}\): \( \mathbf{a} \cdot \mathbf{b} = -1(4) + (-2)(8) + 2(-8) = -4 -16 -16 = -36 \) Since the dot product is not equal to zero, the vectors are neither orthogonal nor parallel. Pair 2: Neither

Step 3: Pair 3: \(\mathbf{a}=\langle -1,-2,2\rangle \) and \(\mathbf{b}=\langle 2,4,-5\rangle \)

Step 1. Check for parallel vectors Verify if vector \(\mathbf{a}\) is a multiple of vector \(\mathbf{b}\) or vice versa: \[ \begin{array}{l} \mathbf{a}=\alpha \mathbf{b} \\ \langle -1,-2,2\rangle = \alpha \langle 2,4,-5\rangle \end{array} \] We can see that \(\mathbf{a}\) is not a multiple of \(\mathbf{b}\), so they are not parallel. Step 2. Check for orthogonal vectors Calculate the dot product of \(\mathbf{a}\) and \(\mathbf{b}\): \( \mathbf{a} \cdot \mathbf{b} = -1(2) + (-2)(4) + 2(-5) = -2 -8 -10 = -20 \) Since the dot product is not equal to zero, the vectors are neither orthogonal nor parallel. Pair 3: Neither So the pairs of vectors are: 1. Orthogonal 2. Neither 3. Neither

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in the study of vectors which tells us about the directional relationship between two vectors. It is calculated by multiplying corresponding components of two vectors and then summing those products. The formula for the dot product of two vectors \( \mathbf{a} \), with components \( \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} \), with components \( \langle b_1, b_2, b_3 \rangle \) is:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
This calculation has important geometric implications. If the dot product of two vectors is zero, it indicates that they are orthogonal, meaning they are at a 90-degree angle to each other. One can easily mistake the absence of a scalar multiple between vectors for orthogonality; however, a zero dot product is the definitive test for this condition.
In the context of the provided exercise, the dot product is used to determine if pairs of vectors are orthogonal. For example, when the dot product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) was calculated and found to be zero, it was concluded that the vectors are orthogonal—as seen with the first pair in the exercise. Understanding the dot product is essential, not just for determining orthogonality, but also for comprehending the angle between vectors and projecting one vector onto another.

Vector Multiplication

In the realm of vectors, vector multiplication can take various forms, with the dot product being one of them. However, another important type of vector multiplication is the cross product, which is distinct from the dot product. The cross product of two vectors results in a third vector that is perpendicular to the plane containing the original two vectors. The cross product is crucial in physics and engineering applications, such as calculating the torque of a force.
While the exercise in question utilizes the dot product, it is helpful for students to distinguish it from the cross product and understand its uniqueness in determining parallelism and orthogonality between vectors. For instance, parallel vectors can be identified when one vector is simply a scalar multiple of another, which is a form of vector multiplication. If no scalar multiple can be found that turns one vector into the other, as shown in all pairs of vectors in our exercise, then the vectors are not parallel.
It's important to note that when identifying parallelism, simple proportionality across corresponding components is what we check for, while orthogonality requires a deeper investigation via the dot product. Keep in mind that in vector multiplication, we're not just multiplying magnitudes; we're considering direction, which is integral to the nature of vector operations.

Multivariable Calculus

The study of functions with more than one variable falls under multivariable calculus. It extends the principles of calculus to functions of several variables. Concepts such as partial derivatives and multiple integrals allow us to analyze and understand how functions change in a space with multiple dimensions. For example, while a single-variable function has a derivative that represents the slope of the curve at a point, a multivariable function has partial derivatives that represent the rate of change along each axis in a multidimensional space.
In relation to vectors, multivariable calculus often involves work with vector fields, where each point in space is assigned a vector. Techniques like gradient, divergence, and curl provide insight into the nature of the vector field such as the direction and magnitude of the steepest ascent, the tendency of the field to diverge from or converge to a point, and the extent of rotation around a point, respectively.
Our exercise, involving vector parallelism and orthogonality, is a practical application of multivariable calculus. The vectors exist in three-dimensional space, and the concepts of dot product and parallelism are directly applicable. Should these vectors represent gradients of functions, orthogonality would indicate that the functions' rates of change are uncorrelated in the direction of the vectors. The principles of multivariable calculus not only provide the tools to perform these calculations but also the context to understand them in a broader scientific and mathematical environment.