Problem 33 - Thomas Calculus

Question

Cancellation in cross products 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

Yes, \( \mathbf{v} = \mathbf{w} \) if \( \mathbf{u} \neq \mathbf{0} \).

Step 1: Understanding the Cross Product Equation

We are given that \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \). This means that the cross product of vector \( \mathbf{u} \) with vector \( \mathbf{v} \) is the same as the cross product with vector \( \mathbf{w} \).

Step 2: Use Property of Cross Product

One of the properties of the cross product states that if \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \), then \( \mathbf{u} \times (\mathbf{v} - \mathbf{w}) = \mathbf{0} \).

Step 3: Analyzing the Zero Cross Product

The equation \( \mathbf{u} \times (\mathbf{v} - \mathbf{w}) = \mathbf{0} \) implies that either \( \mathbf{v} - \mathbf{w} \) is a vector parallel to \( \mathbf{u} \) or it is the zero vector itself. This is because the cross product of two vectors is zero if the vectors are parallel or one of them is the zero vector.

Step 4: Analyze \( \mathbf{u} \neq \mathbf{0} \) Condition

Given that \( \mathbf{u} eq \mathbf{0} \), the zero condition must apply to the vector \( \mathbf{v} - \mathbf{w} \). Thus, \( \mathbf{v} - \mathbf{w} \) could simply be the zero vector itself, implying that \( \mathbf{v} = \mathbf{w} \).

Step 5: Conclusion

With the understanding from the properties and conditions above, if \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \) and \( \mathbf{u} eq \mathbf{0} \), it indeed implies that \( \mathbf{v} = \mathbf{w} \).

Key Concepts

Vector Algebra

Vector algebra is a fundamental mathematical framework used when dealing with quantities that have both a magnitude and a direction, such as vectors. Vectors are crucial in many fields like physics, engineering, and computer graphics.
We use vectors to add, subtract, and multiply in various ways, giving us insights into spatial relationships and physical phenomena.

One common operation is the cross product, symbolized as \( \mathbf{u} \times \mathbf{v} \), which involves two vectors and produces another vector perpendicular to both. In vector algebra, understanding how such operations work helps in solving complex problems like the given exercise, where the identity \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \) leads us to conclude that \( \mathbf{v} = \mathbf{w} \) when \( \mathbf{u} eq \mathbf{0} \). This is just one application of vector algebra, showing how critical it is to mastery in solving vector-based problems.

Cross Product Properties

The cross product is a specific vector operation used to find a vector that's perpendicular to two given vectors. Understanding its properties is essential when dealing with vector equations.

• If given vectors \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \), the cross product's properties help us manipulate and solve equations involving these vectors.
• For example, the property \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \) implies that \( \mathbf{u} \times (\mathbf{v} - \mathbf{w}) = \mathbf{0} \).
• This condition tells us about the relationship between \( \mathbf{v} \) and \( \mathbf{w} \); specifically, if this equation holds and \( \mathbf{u} eq \mathbf{0} \), \( \mathbf{v} \) and \( \mathbf{w} \) must be equal.

These properties make the cross product a powerful tool in problem-solving scenarios where vector equality needs to be demonstrated or verified.

Zero Vector Condition

In vector algebra, the zero vector condition is a powerful concept that can simplify complex vector equations. Understanding this is key when analyzing vector equations like \( \mathbf{u} \times (\mathbf{v} - \mathbf{w}) = \mathbf{0} \).

• This equation suggests that the resulting vector from the cross product operation is the zero vector.
• The zero vector is unique because it has no magnitude or direction, acting as an "identity" in vector spaces.
• For the above equation to hold, either \( \mathbf{v} - \mathbf{w} \) is parallel to \( \mathbf{u} \) or \( \mathbf{v} - \mathbf{w} \) itself is zero.
• Since \( \mathbf{u} eq \mathbf{0} \) is given, \( \mathbf{v} - \mathbf{w} \) must be the zero vector, meaning \( \mathbf{v} = \mathbf{w} \).

With this condition in mind, one can better navigate and resolve equations involving vectors and their interactions, demonstrating their parallel or identical nature as needed.