Problem 33 - University Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

No, \( \mathbf{v} \) and \( \mathbf{w} \) can differ by a vector parallel to \( \mathbf{u} \).

Step 1: Understanding the Cross Product Equation

We are given that \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \). The cross product \( \mathbf{u} \times \mathbf{v} \) results in a vector that is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \). Due to the given equation, both \( \mathbf{u} \times \mathbf{v} \) and \( \mathbf{u} \times \mathbf{w} \) result in the same vector that is perpendicular to \( \mathbf{u} \).

Step 2: Analyzing the Implication of the Equation

Since the resulting cross products are equal and \( \mathbf{u} eq \mathbf{0} \), it implies that \( \mathbf{u} \) is not collinear with either \( \mathbf{v} \) or \( \mathbf{w} \). However, it means \( \mathbf{v} - \mathbf{w} \) must be collinear with \( \mathbf{u} \) because their cross product with \( \mathbf{u} \) results in the zero vector.

Step 3: Applying the Cross Product Properties

Using the property \( \mathbf{u} \times (\mathbf{v} - \mathbf{w}) = \mathbf{0} \), since \( \mathbf{u} \times \mathbf{0} = \mathbf{0} \), \( \mathbf{u} \) and \( \mathbf{v} - \mathbf{w} \) must be parallel vectors. This indicates that \( \mathbf{v} - \mathbf{w} = k\mathbf{u} \) where \( k \) is a scalar.

Step 4: Conclusion About \( \mathbf{v} \) and \( \mathbf{w} \)

Since \( \mathbf{v} - \mathbf{w} = k\mathbf{u} \), \( \mathbf{v} \) and \( \mathbf{w} \) can be different if \( k eq 0 \). Thus, the statement \( \mathbf{v} = \mathbf{w} \) is not necessarily true, as they can differ by a component in the direction of \( \mathbf{u} \).

Key Concepts

Cross Product

In vector algebra, the cross product is a fundamental operation for determining the relationship between two vectors. The cross product of two vectors, \( \mathbf{u} \times \mathbf{v} \), results in a third vector that is perpendicular to the plane formed by these vectors. This perpendicular vector is unique in its direction and magnitude. The magnitude of the cross product is given by the formula:

• \( |\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin \theta \)

where \( \theta \) is the angle between the original vectors \( \mathbf{u} \) and \( \mathbf{v} \). The direction of the cross product vector follows the right-hand rule: if one points the index finger along \( \mathbf{u} \) and the middle finger along \( \mathbf{v} \), the thumb points in the direction of the cross product. This operation highlights the significance of orientation since the order of the vectors affects the direction of the resultant vector.
Cross product is only applicable in three-dimensional space, which makes it a powerful tool in physics and engineering for understanding rotational forces and integrals in three dimensions.

Non-Collinear Vectors

Non-collinear vectors are vectors that are not lying on the same line. In other words, their direction does not merely scale each other. If vectors are collinear, their cross product is zero, because they do not span a plane. In the case of non-collinear vectors, their cross product will result in a non-zero vector, pointing away from the plane they form.
This property is crucial for understanding the exercise problem. Given \( \mathbf{u} eq \mathbf{0} \), it is implied that \( \mathbf{u} \) itself is not collinear with the results of the cross product operations \( \mathbf{u} \times \mathbf{v} \) and \( \mathbf{u} \times \mathbf{w} \). This ensures that \( \mathbf{v} \) and \( \mathbf{w} \) must be non-collinear relative to \( \mathbf{u} \).
Understanding non-collinear vectors is foundational for solving problems involving orientations and spaces defined by multiple vectors, which are common in geometry and physics.

Vector Equation Analysis

Vector equation analysis allows us to understand relationships and dependencies between vectors based on operations such as addition or cross products. In the context of the given exercise, analyzing the equation \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \) provides insight into the relationship between vectors \( \mathbf{v} \) and \( \mathbf{w} \).
By rearranging this equation into \( \mathbf{u} \times (\mathbf{v} - \mathbf{w}) = \mathbf{0} \), we apply the property that the cross product of any vector with a zero vector is zero. This indicates that the vector \( \mathbf{v} - \mathbf{w} \) is parallel to \( \mathbf{u} \), expressed as \( \mathbf{v} - \mathbf{w} = k\mathbf{u} \) where \( k \) is some scalar.

• If \( k = 0 \), it implies \( \mathbf{v} = \mathbf{w} \).
• If \( k eq 0 \), \( \mathbf{v} \) and \( \mathbf{w} \) could be different by a magnitude that is a scalar multiple of \( \mathbf{u} \).

This analysis enhances understanding of vector behavior in spaces and reveals whether vectors are merely scaled versions of others within a directional space, critical for higher-dimensional algebra and calculus.