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Chapter 11: Problem 4

If \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, what is the magnitude of \(\mathbf{u} \times \mathbf{v} ?\)

Short Answer

Expert verified
Answer: The magnitude of the cross product between two orthogonal vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by \(\|\mathbf{u} \times \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 \|\mathbf{v}\|^2}\).

Step by step solution

01

Write down the relationship between the magnitudes of cross and dot products

Recall that for any two vectors \(\mathbf{u}\) and \(\mathbf{v}\), the relationship between the magnitudes of their cross and dot products can be written as: $$\|\mathbf{u} \times \mathbf{v}\|^2 = \|\mathbf{u}\|^2 \|\mathbf{v}\|^2 - (\mathbf{u} \cdot \mathbf{v})^2$$
02

Use the fact that \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal

Since \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, their dot product is zero, i.e., \(\mathbf{u} \cdot \mathbf{v} = 0\). We can now use this fact in the relationship we derived in step 1: $$\|\mathbf{u} \times \mathbf{v}\|^2 = \|\mathbf{u}\|^2 \|\mathbf{v}\|^2 - (0)^2$$
03

Rewrite the expression for the magnitude of their cross product

Simplify the above expression as follows: $$\|\mathbf{u} \times \mathbf{v}\|^2 = \|\mathbf{u}\|^2 \|\mathbf{v}\|^2 $$ Now, to find the magnitude of the cross product, we take the square root of both sides: $$\|\mathbf{u} \times \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 \|\mathbf{v}\|^2}$$
04

Final answer

The magnitude of the cross product between two orthogonal vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by: $$\|\mathbf{u} \times \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 \|\mathbf{v}\|^2}$$

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