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Chapter 12: Problem 10

Sketch the following vectors \(\mathbf{u}\) and \(\mathbf{v} .\) Then compute \(|\mathbf{u} \times \mathbf{v}|\) and show the cross product on your sketch. $$\mathbf{u}=\langle 0,4,0\rangle, \mathbf{v}=\langle 0,0,-8\rangle$$

Short Answer

Expert verified
Question: Represent vector u with components (0,4,0) and vector v with components (0,0,-8) graphically. Compute the cross product of u and v, including its magnitude, and represent it on the sketch. Answer: After sketching vectors u, v, and their cross product, u x v, we find that u x v = <-32, 0, 0> and its magnitude is |u x v| = 32.

Step by step solution

01

Sketch the Vectors \(\mathbf{u}\) and \(\mathbf{v}\)

Plot the two given vectors in 3D cartesian coordinate system. Vector \(\mathbf{u}\) has its initial point at the origin and terminal point at \((0,4,0)\). Similarly, vector \(\mathbf{v}\) has its initial point at the origin and terminal point at \((0,0,-8)\).
02

Compute the Cross Product \(\mathbf{u} \times \mathbf{v}\)

To compute the cross product of vectors \(\mathbf{u}\) and \(\mathbf{v}\), we use the formula: $$\mathbf{u} \times \mathbf{v} = \langle u_{2}v_{3}-u_{3}v_{2}, u_{3}v_{1}-u_{1}v_{3}, u_{1}v_{2}-u_{2}v_{1}\rangle.$$ Plugging in the components of vectors \(\mathbf{u}\) and \(\mathbf{v}\), we get: $$\mathbf{u} \times \mathbf{v} = \langle (4)(-8)-(0)(0), (0)(0)-(0)(-8), (0)(0)-(4)(0) \rangle = \langle -32,0,0 \rangle.$$
03

Compute the Magnitude \(|\mathbf{u} \times \mathbf{v}|\)

To compute the magnitude of the cross product, we use the formula: $$|\mathbf{u} \times \mathbf{v}| = \sqrt{(-32)^2 + 0^2 + 0^2} = \sqrt{1024} = 32.$$
04

Sketch the Cross Product \(\mathbf{u} \times \mathbf{v}\)

As we calculated the cross product \(\mathbf{u} \times \mathbf{v} =\langle -32, 0, 0 \rangle\), this vector has its initial point at the origin and terminal point at \((-32, 0, 0)\). Add this vector to your existing sketch having vectors \(\mathbf{u}\) and \(\mathbf{v}\). The sketch now should have vectors \(\mathbf{u}\), \(\mathbf{v}\), and their cross product \(\mathbf{u} \times \mathbf{v}\) with magnitudes \(|u|=4\), \(|v|=8\), and \(|\mathbf{u} \times \mathbf{v}|=32\).

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Most popular questions from this chapter

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