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Chapter 12: Problem 1

Explain how to find the magnitude of the cross product \(\mathbf{u} \times \mathbf{v}\)

Short Answer

Expert verified
Answer: To find the magnitude of the cross product of two vectors, first calculate their cross product using the formula \(\mathbf{u} \times \mathbf{v} = \langle (u_2 v_3 - u_3 v_2), (u_3 v_1 - u_1 v_3), (u_1 v_2 - u_2 v_1) \rangle\), and then find its magnitude using the formula \(||\mathbf{u} \times \mathbf{v}|| = \sqrt{(u_2 v_3 - u_3 v_2)^2 + (u_3 v_1 - u_1 v_3)^2 + (u_1 v_2 - u_2 v_1)^2}\).

Step by step solution

01

Understanding the cross product

The cross product, denoted by \(\times\), is a binary operation that takes two vectors as inputs and returns a new vector. The cross product is only defined for three-dimensional vectors and has the property that the resulting vector is orthogonal (perpendicular) to both input vectors. The magnitude of the cross product can be thought of as the area of the parallelogram formed by the two vectors.
02

Formula for the cross product

Given two vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\), their cross product is defined as: \(\mathbf{u} \times \mathbf{v} = \langle (u_2 v_3 - u_3 v_2), (u_3 v_1 - u_1 v_3), (u_1 v_2 - u_2 v_1) \rangle\)
03

Formula for the magnitude of a vector

The magnitude of a vector \(\mathbf{w} = \langle w_1, w_2, w_3 \rangle\) is defined as: \(||\mathbf{w}|| = \sqrt{w_1^2 + w_2^2 + w_3^2}\)
04

Calculate the cross product

Using the formula provided in step 2, compute the cross product \(\mathbf{u} \times \mathbf{v}\).
05

Find the magnitude of the cross product

Now that we have the cross product, calculate its magnitude using the formula provided in step 3: \(||\mathbf{u} \times \mathbf{v}|| = \sqrt{(u_2 v_3 - u_3 v_2)^2 + (u_3 v_1 - u_1 v_3)^2 + (u_1 v_2 - u_2 v_1)^2}\) Following these steps will give you the magnitude of the cross product of the given vectors.

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Most popular questions from this chapter

Determine whether the following statements are true and give an explanation or counterexample. a. The line \(\mathbf{r}(t)=\langle 3,-1,4\rangle+t\langle 6,-2,8\rangle\) passes through the origin. b. Any two nonparallel lines in \(\mathbb{R}^{3}\) intersect. c. The curve \(\mathbf{r}(t)=\left\langle e^{-t}, \sin t,-\cos t\right\rangle\) approaches a circle as \(t \rightarrow \infty\). d. If \(\mathbf{r}(t)=e^{-t^{2}}\langle 1,1,1\rangle\) then \(\lim _{t \rightarrow \infty} \mathbf{r}(t)=\lim _{t \rightarrow-\infty} \mathbf{r}(t)\).

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