Problem 34 - Thomas Calculus

Question

Double cancellation 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, \( \mathbf{v} = \mathbf{w} \); both conditions imply \( \mathbf{v} - \mathbf{w} = \mathbf{0} \).

Step 1: Understand the Given Condition

We are provided with two vector conditions: 1. A cross product condition: \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \) 2. A dot product condition: \( \mathbf{u} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{w} \)The vector \( \mathbf{u} \) is not the zero vector. Our goal is to determine if these conditions imply that \( \mathbf{v} = \mathbf{w} \).

Step 2: Analyze the Cross Product Condition

The cross product condition \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \) implies that the vectors \( \mathbf{v} - \mathbf{w} \) must be parallel to \( \mathbf{u} \). This is because the cross product of two vectors equals zero when they are scalar multiples of each other, indicating collinearity.

Step 3: Analyze the Dot Product Condition

The dot product condition \( \mathbf{u} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{w} \) translates to \( \mathbf{u} \cdot (\mathbf{v} - \mathbf{w}) = 0 \). This equation implies that the vector \( \mathbf{v} - \mathbf{w} \) is orthogonal to \( \mathbf{u} \).

Step 4: Examine Both Conditions

For the vector \( \mathbf{v} - \mathbf{w} \) to satisfy both conditions - being parallel to and orthogonal to \( \mathbf{u} \), it can only be the zero vector. If \( \mathbf{v} - \mathbf{w} = \mathbf{0} \), then \( \mathbf{v} = \mathbf{w} \).

Step 5: Conclusion

Given \( \mathbf{u} eq \mathbf{0} \), both conditions together imply that \( \mathbf{v} - \mathbf{w} \) must be the zero vector for it to be both parallel and orthogonal to \( \mathbf{u} \). Therefore, \( \mathbf{v} = \mathbf{w} \).

Key Concepts

Cross Product

The cross product is a mathematical operation that takes two vectors and returns another vector that is perpendicular to both of them. It is represented as \( \mathbf{u} \times \mathbf{v} \) and is widely used in physics and engineering to determine the vector orthogonal to a plane.

• The result of the cross product is a vector, not a scalar, and has a direction determined by the right-hand rule.
• The magnitude of the cross product reflects the area of the parallelogram formed by the two vectors.

In the given exercise, the cross product condition \( \mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} \) indicates that the vector difference \( \mathbf{v} - \mathbf{w} \) is parallel to \( \mathbf{u} \). This concept helps identify whether vectors are collinear (lie along the same line).

Dot Product

The dot product, also known as the scalar product, involves two vectors and results in a scalar quantity. It is expressed as \( \mathbf{u} \cdot \mathbf{v} \) and is used to project one vector onto another.

• The dot product gives insight into how much one vector "goes in the direction" of another.
• A dot product of zero implies that the vectors are orthogonal, or at right angles to each other.

In the scenario of the exercise, the condition \( \mathbf{u} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{w} \) implies that the vector \( \mathbf{v} - \mathbf{w} \) is orthogonal to \( \mathbf{u} \). Here, it helps identify when two vectors are at right angles with respect to another vector.

Zero Vector

The zero vector, often denoted as \( \mathbf{0} \), is a unique vector that has zero magnitude and is directionless. It serves as an additive identity in vector algebra, meaning any vector added to the zero vector remains unchanged.

• The zero vector is crucial in vector mathematics, symbolizing the concept of no movement or change in both position and magnitude.

In our exercise, the idea that the vector \( \mathbf{v} - \mathbf{w} \) is the zero vector (after fulfilling both cross and dot product conditions) confirms that \( \mathbf{v} = \mathbf{w} \). This means the vectors must be identical for all conditions to hold, highlighting the zero vector's role in proving equality.

Parallel Vectors

Parallel vectors are vectors that have the same or exact opposite direction. They can be any magnitude but must maintain the same direction. When vectors are parallel, they are scalar multiples of each other.

• If \( \mathbf{a} \) and \( \mathbf{b} \) are parallel, then there exists a scalar \( k \) such that \( \mathbf{a} = k \mathbf{b} \).

In the context of our exercise, when the cross product \( \mathbf{u} \times (\mathbf{v} - \mathbf{w}) = \mathbf{0} \), it implies that the vector \( \mathbf{v} - \mathbf{w} \) is parallel to vector \( \mathbf{u} \). This is a critical concept when determining that vectors lie along the same line or plane, aiding in vector comparison and analysis.