Problem 36 - Essential Calculus Early Transcendentals

Question

Suppose that a and b are nonzero vectors. (a) Under what circumstances is \(\operatorname{comp}_{\mathbf{a}} \mathbf{b}=\operatorname{comp}_{\mathbf{b}} \mathbf{a} ?\) (b) Under what circumstances is \(\operatorname{proj}_{a} b=\operatorname{proj}_{b} a ?\)

Step-by-Step Solution

Answer

(a) \( \|\mathbf{a}\| = \|\mathbf{b}\| \); (b) \( \mathbf{a} = \pm \mathbf{b} \).

Step 1: Understanding the Components of a Vector

The component of vector \( \mathbf{b} \) in the direction of vector \( \mathbf{a} \) is given by \( \operatorname{comp}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|} \). Similarly, the component of \( \mathbf{a} \) in the direction of \( \mathbf{b} \) is \( \operatorname{comp}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|} \).

Step 2: Equating the Components

For these components to be equal, \( \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|} \). This implies \( \|\mathbf{a}\| = \|\mathbf{b}\| \) since \( \mathbf{a} \cdot \mathbf{b} eq 0 \) if \( \mathbf{a} \) and \( \mathbf{b} \) are nonzero and not orthogonal.

Step 3: Understanding the Projections of a Vector

The projection of vector \( \mathbf{b} \) on \( \mathbf{a} \) is \( \operatorname{proj}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|^2} \mathbf{a} \). Similarly, \( \operatorname{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|^2} \mathbf{b} \).

Step 4: Equating the Projections

To have \( \operatorname{proj}_{\mathbf{a}} \mathbf{b} = \operatorname{proj}_{\mathbf{b}} \mathbf{a} \), we must have \( \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|^2} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|^2} \mathbf{b} \). This implies \( \|\mathbf{a}\|\mathbf{a} = \|\mathbf{b}\|\mathbf{b} \), meaning either \( \mathbf{a} = \mathbf{b} \) or \( \mathbf{a} = -\mathbf{b} \).

Step 5: Solution to the Components Circumstance

The components \( \operatorname{comp}_{\mathbf{a}} \mathbf{b} \) and \( \operatorname{comp}_{\mathbf{b}} \mathbf{a} \) are equal when the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \) are the same.

Step 6: Solution to the Projections Circumstance

The projections \( \operatorname{proj}_{\mathbf{a}} \mathbf{b} \) and \( \operatorname{proj}_{\mathbf{b}} \mathbf{a} \) are equal when \( \mathbf{a} = \mathbf{b} \) or \( \mathbf{a} = -\mathbf{b} \).

Key Concepts

Vector Components

When discussing vectors in mathematics, components are a fundamental concept that helps us understand how vectors interact with each other. Specifically, the component of one vector in the direction of another gives us an idea of how much one vector extends in the direction of the other. This is crucial when analyzing vector behavior in physics and engineering.

To find the component of vector \( \mathbf{b} \) in the direction of vector \( \mathbf{a} \), we use the formula:

• \( \operatorname{comp}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|} \).

Similarly, the component of \( \mathbf{a} \) in the direction of \( \mathbf{b} \) is:

• \( \operatorname{comp}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|} \).

These equations show us how the angle and magnitude of the vectors affect how they project onto each other. The components become equal when the magnitudes of vectors \( \mathbf{a} \) and \( \mathbf{b} \) are the same.

Vector Projection

Vector projection is an important operation that lets us express one vector completely in terms of another. It is often used to simplify problems in physics and computer graphics by projecting a vector onto another to see how much of it lies in that direction.

The projection of vector \( \mathbf{b} \) onto vector \( \mathbf{a} \) is given by:

• \( \operatorname{proj}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|^2} \mathbf{a} \).

This formula shows that the projection is in the direction of \( \mathbf{a} \) and scaled by how much \( \mathbf{b} \) is aligned with \( \mathbf{a} \).

An important scenario arises when we equate the projections of the vectors on each other. For \( \operatorname{proj}_{\mathbf{a}} \mathbf{b} \) to equal \( \operatorname{proj}_{\mathbf{b}} \mathbf{a} \), the vectors must be parallel or anti-parallel—meaning \( \mathbf{a} = \mathbf{b} \) or \( \mathbf{a} = -\mathbf{b} \).

Dot Product

The dot product is a primary operation in vector mathematics that provides a scalar result representing the product of two vectors' magnitudes and the cosine of the angle between them. It is essential in determining both the direction and magnitude of one vector relative to another.

The dot product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by:

• \( \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta \),

where \( \theta \) is the angle between the vectors.

This product is pivotal in calculations involving vector components and projections, helping to determine the extent to which vectors point in the same direction. It also plays a key role in deriving equations like those used for vector components and projections.

Vector Magnitude

Understanding vector magnitude is crucial because it gives information about the size or length of the vector, irrespective of its direction. In many problems, especially in physics and engineering, the magnitude is more of interest than the direction.

The magnitude of a vector \( \mathbf{a} \) is calculated as:

• \( \|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2 + a_3^2} \),

where \( a_1, a_2, \) and \( a_3 \) are the components of the vector.

Having equal magnitudes, as required for equal vector components, leads to interesting geometric interpretations, such as the vectors being aligned in size but not necessarily in direction. Understanding magnitudes aids in solving equations where vectors interact physically, such as determining force, velocity, or displacement.