   Chapter 12, Problem 7RQ

Chapter
Section
Textbook Problem

# Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.7. For any vectors u and v in V3, |u × v| = |v × u|.

To determine

Whether the statement |u×v|=|v×u| is true or false.

Explanation

Formula used:

Consider the two three-dimensional vectors a=a1,a2,a3 and b=b1,b2,b3.

Cross product of vectors:

Write the expression for the cross product of vectors a and b (a×b).

a×b=|ijka1a2a3b1b2b3|

a×b=(a2b3b2a3)i(a1b3b1a3)j+(a1b2b1a2)k (1)

Write the expression for the magnitude of vector a (|a|).

|a|=a12+a22+a32 (2)

Consider the given statement.

|u×v|=|v×u|

Consider two vectors, u=u1,u2,u3 and v=v1,v2,v3.

Find the cross product of u×v by using equation (1).

u×v=|ijku1u2u3v1v2v3|=(u2v3v2u3)i(u1v3v1u3)j+(u1v2v1u2)k

Find the magnitude of u×v (|u×v|) by using equation (2).

|u×v|=(u2v3v2u3)2+((u1v3v1u3))2+(u1v2v1u2)2

|u×v|=(u2v3v2u3)2+(u1v3v1u3)2+(u1v2v1u2)2 (3)

Find the cross product of v×u by the use of equation (1)

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