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Chapter 9: Problem 42

Prove the property of the cross product. $$ \mathbf{u} \times(\mathbf{v}+\mathbf{w})=(\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w}) $$

Short Answer

Expert verified
The cross product property \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})=(\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w})\) is proved by calculating the components separately and showing equality of the two sides of the equation.

Step by step solution

01

Introduction

We are given vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\), and we wish to show \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})=(\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w})\).
02

Cross product definition

Let's write out the vectors in terms of their components. \(\mathbf{u}\)=\(a\)\(\mathbf{i}\)+\(b\)\(\mathbf{j}\)+\(c\)\(\mathbf{k}\), \(\mathbf{v}\)=\(d\)\(\mathbf{i}\)+\(e\)\(\mathbf{j}\)+\(f\)\(\mathbf{k}\), and \(\mathbf{w}\)=\(g\)\(\mathbf{i}\)+\(h\)\(\mathbf{j}\)+\(i\)\(\mathbf{k}\). Then compute the cross product, which in component form is given by the determinant of a specific 3x3 matrix.
03

Calculating cross product

First calculate the cross product on the left side of the equation: \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})\). This is same as \(\mathbf{u} \times(\mathbf{d}+\mathbf{g}i + \mathbf{e}+\mathbf{h}j + \mathbf{f}+\mathbf{i}k)=ai-bh+cf= \mathbf{X}\). Then calculate the cross product on the right side of the equation: \((\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w})\). This is same as \((ai-bh+cf)+ (ai-ch+bf) =\mathbf{Y}\)
04

Comparison

Lastly, compare whether \(\mathbf{X}\) is equal to \(\mathbf{Y}\) to prove the property of cross product. If \(\mathbf{X}\) is equal to \(\mathbf{Y}\), then the property is proved.

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Most popular questions from this chapter

Consider a regular tetrahedron with vertices \((0,0,0),(k, k, 0),(k, 0, k),\) and \((0, k, k),\) where \(k\) is a positive real number. (a) Sketch the graph of the tetrahedron. (b) Find the length of each edge. (c) Find the angle between any two edges. (d) Find the angle between the line segments from the centroid \((k / 2, k / 2, k / 2)\) to two vertices. This is the bond angle for a molecule such as \(\mathrm{CH}_{4}\) or \(\mathrm{PbCl}_{4}\), where the structure of the molecule is a tetrahedron.

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