Problem 34 - University Calculus: Early Transcendentals

Question

If \(\mathbf{u} \neq \mathbf{0}\) and if \(\mathbf{u} \times \mathbf{v}=\mathbf{u} \times \mathbf{w}\) and \(\mathbf{u} \cdot \mathbf{v}=\mathbf{u} \cdot \mathbf{w},\) then does \(\mathbf{v}=\mathbf{w} ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

Yes, \( \mathbf{v} = \mathbf{w} \) because both cross and dot product conditions imply they are the same with respect to \( \mathbf{u} \).

Step 1: Analyze the Cross Product Condition

The given condition \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \) implies that the cross products of \( \mathbf{u} \) with \( \mathbf{v} \) and \( \mathbf{w} \) are equal. For the cross product \( \mathbf{u} \times \mathbf{x} \) to be zero (and hence equal), \( \mathbf{x} \) must be a linear combination of \( \mathbf{u} \). Since \( \mathbf{u} eq \mathbf{0} \), this suggests \( \mathbf{v} \) and \( \mathbf{w} \) lie in the same plane as \( \mathbf{u} \).

Step 2: Analyze the Dot Product Condition

The condition \( \mathbf{u} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{w} \) indicates that the projections of \( \mathbf{v} \) and \( \mathbf{w} \) in the direction of \( \mathbf{u} \) are equal. This signifies that the components of \( \mathbf{v} \) and \( \mathbf{w} \) along the direction of \( \mathbf{u} \) are identical.

Step 3: Combine Information from Both Conditions

Given that both the cross product and dot product conditions are satisfied, vectors \( \mathbf{v} \) and \( \mathbf{w} \) are constrained in their relationship with \( \mathbf{u} \). The dot product condition ensures their scalar projections on \( \mathbf{u} \) are equal, and the cross product condition ensures they lie in the same direction with respect to the vector \( \mathbf{u} \). This combination implies that \( \mathbf{v} = \mathbf{w} \), since there is no other vector with distinct directionality that would satisfy both conditions simultaneously, except by being equal.

Key Concepts

Cross Product

The cross product of two vectors \(\mathbf{a} \) and \(\mathbf{b} \), denoted \(\mathbf{a} \times \mathbf{b} \), results in a third vector that is orthogonal to the plane formed by \(\mathbf{a} \) and \(\mathbf{b}\). This operation can be used to determine the "turned direction" of two vectors with respect to one another.

• The magnitude of the cross product is given by \( |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin(\theta)\), where \(\theta\) is the angle between \(\mathbf{a}\) and \(\mathbf{b}\).
• If \(\mathbf{a} \times \mathbf{b} = \mathbf{0}\), then \(\mathbf{a}\) and \(\mathbf{b}\) are parallel (which includes the case when they are both zero vectors).

In the given problem, the condition \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \) implies that the vectors \(\mathbf{v}\) and \(\mathbf{w}\) must lie on the same plane as \(\mathbf{u}\). They are not independent and have the same relative orientation with \(\mathbf{u}\). Thus, leading to the conclusion that if their cross products with vector \(\mathbf{u}\) are equal, their directions must also be the same concerning \(\mathbf{u}\).

Dot Product

The dot product, or scalar product, measures how much two vectors "work together" in terms of their alignment. This is represented by the formula \(\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\theta)\) where \(\theta\) is the angle between them.

• The result of a dot product is a scalar, not a vector.
• If \(\mathbf{a} \cdot \mathbf{b} = 0\), then \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal, or perpendicular to each other.

In the exercise, considering \( \mathbf{u} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{w} \), the vectors \(\mathbf{v}\) and \(\mathbf{w}\) have the same magnitude of projection along the direction of \(\mathbf{u}\). This finding tells us that, although neither the vectors \(\mathbf{v}\) and \(\mathbf{w}\) are necessarily identical in magnitude, their influence in the direction of \(\mathbf{u}\) is identical. Together with the cross product condition, these vectors must essentially mirror each other in direction and magnitude, leading to the conclusion that they are equal vectors.

Linear Combination

A linear combination involves constructing a new vector by using a set of original vectors, each multiplied by a corresponding scalar. Given vectors \(\mathbf{a}\) and \(\mathbf{b}\), a linear combination would have the form \( c_1\mathbf{a} + c_2\mathbf{b} \), where \(c_1\) and \(c_2\) are scalars.

• Vectors lying in the same plane as a given vector \(\mathbf{u}\) mean that they can be expressed as a linear combination of vector \(\mathbf{u}\) and any other vector in the same plane.
• For example, if a vector \(\mathbf{x}\) is a linear combination of \(\mathbf{u}\), then \(\mathbf{x} = c\mathbf{u}\) for some scalar \(c\).

In this exercise, we utilize the concept that if both \(\mathbf{v}\) and \(\mathbf{w}\) interact with \(\mathbf{u}\) linearly in the same way through their dot and cross product equivalences, then they are indeed linear combinations of \(\mathbf{u}\) that equate, consequently asserting that \(\mathbf{v} = \mathbf{w}\). Therefore, understanding linear combinations is crucial for interpreting vector equality discussions effectively in this context.