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Chapter 9: Problem 48

Prove \(\|\mathbf{u} \times \mathbf{v}\|=\|\mathbf{u}\|\|\mathbf{v}\|\) if \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Short Answer

Expert verified
The magnitude of the cross product of two orthogonal vectors \(\mathbf{u}\) and \(\mathbf{v}\) equals the product of their magnitudes, as defined by the relation \(\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\|\|\mathbf{v}\|\).

Step by step solution

01

Define the terms

First, let's make sure we understand the notation: \(\mathbf{u}\) and \(\mathbf{v}\) are vectors; \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) represent their magnitudes; \(\mathbf{u} \times \mathbf{v}\) is their cross product; and the two-vectors are orthogonal, which means their dot product is zero.
02

Apply the formula for the magnitude of the cross product

The formula for the magnitude (or length) of the cross product of two vectors is given by \(\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\|\|\mathbf{v}\|\sin(\theta)\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\).
03

Utilize the definition of orthogonality

The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, which means \(\theta\) equals 90 degrees. The sine of 90 degrees is 1.
04

Substitution

Substitute \(\sin(90^\circ)\) with 1 in the formula from Step 2. The formula then simplifies to \(\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\|\|\mathbf{v}\|\), thereby proving the given property.

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Most popular questions from this chapter

In Exercises 31 and 32 , find the component of \(u\) that is orthogonal to \(\mathbf{v},\) given \(\mathbf{w}_{\mathbf{1}}=\operatorname{proj}_{\mathbf{v}} \mathbf{u}\). $$ \mathbf{u}=\langle 6,7\rangle, \quad \mathbf{v}=\langle 1,4\rangle, \quad \operatorname{proj}_{\mathbf{v}} \mathbf{u}=\langle 2,8\rangle $$

In Exercises 75 and \(76,\) sketch the vector \(v\) and write its component form. \(\mathbf{v}\) lies in the \(y z\) -plane, has magnitude 2 , and makes an angle of \(30^{\circ}\) with the positive \(y\) -axis.

Find the magnitude of \(v\). \(\mathbf{v}=\langle 1,0,3\rangle\)

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 vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(2 \mathbf{z}-3 \mathbf{u}=\mathbf{w}\)
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