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Chapter 9: Problem 46

Prove the property of the cross product. $$ \mathbf{u} \times \mathbf{v} \text { is orthogonal to both } \mathbf{u} \text { and } \mathbf{v} \text { . } $$

Short Answer

Expert verified
By applying the definitions of cross product and dot product, and the principle of orthogonality, it has been proven that the cross product of any two vectors is orthogonal to both of those vectors.

Step by step solution

01

Definition of cross product

First, recall that the cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) in \( \mathbb{R}^3 \), denoted \( \mathbf{u} \times \mathbf{v} \), is another vector \( \mathbf{w} \) in \( \mathbb{R}^3 \) that is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \), and whose magnitude is equal to the area of the parallelogram spanned by \( \mathbf{u} \) and \( \mathbf{v} \).
02

Orthogonality test

Then, proceed by proving that \( \mathbf{w}=\mathbf{u} \times \mathbf{v} \) is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \). By definition, two non-null vectors are orthogonal if and only if their dot product is zero, i.e., \( \mathbf{a} \cdot \mathbf{b}=0 \). Therefore, for vector \( \mathbf{w} \) to be orthogonal to vectors \(\mathbf{u} \) and \( \mathbf{v} \), it must satisfy the conditions \( \mathbf{w} \cdot \mathbf{u} = 0 \) and \( \mathbf{w} \cdot \mathbf{v} = 0 \).
03

Apply distributive law

Verify these conditions. It's known that cross product is distributive over addition and dot product is distributive over addition. If factored out, we get \( \mathbf{w} \cdot \mathbf{u} = \mathbf{u} \times \mathbf{v} \cdot \mathbf{u} = \mathbf{u} \cdot \mathbf{u} \times \mathbf{v} = 0 \) and \( \mathbf{w} \cdot \mathbf{v} = \mathbf{u} \times \mathbf{v} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{v} \times \mathbf{v} = 0 \). Therefore, \( \mathbf{u} \times \mathbf{v} \) is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \).

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Most popular questions from this chapter

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{array}{l} \mathbf{u}=\langle 2,-3,1\rangle \\ \mathbf{v}=\langle-1,-1,-1\rangle \end{array} $$

In Exercises 33-36, (a) find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\), and (b) find the vector component of u orthogonal to v. $$ \mathbf{u}=\langle 2,3\rangle, \quad \mathbf{v}=\langle 5,1\rangle $$

In Exercises 49 and \(50,\) find each scalar multiple of \(v\) and sketch its graph. \(\mathbf{v}=\langle 1,2,2\rangle\) (a) \(2 \mathbf{v}\) (b) \(-\mathbf{v}\) (c) \(\frac{3}{2} \mathbf{v}\) (d) \(0 \mathbf{v}\)

Find the component form and magnitude of the vector \(u\) with the given initial and terminal points. Then find a unit vector in the direction of \(\mathbf{u}\). \(\frac{\text { Initial Point }}{(-4,3,1)}\) \(\frac{\text { Terminal Point }}{(-5,3,0)}\)

In Exercises 47 and \(48,\) the vector \(v\) and its initial point are given. Find the terminal point. \(\mathbf{v}=\langle 3,-5,6\rangle\) Initial point: (0,6,2)
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