   Chapter 12.4, Problem 37E

Chapter
Section
Textbook Problem

# Use the scalar triple product to verify that the vectors u = i + 5j −2k, v = 3i − j, and w = 5i + 9j − 4k are coplanar.

To determine

To verify: The vectors u=i+5j2k,v=3ij and w=5i+9j4k are coplanar.

Explanation

Given:

u=i+5j2k,v=3ij and w=5i+9j4k .

Formula used:

Write the expression for cross product between a and b vectors.

a×b=|ijka1a2a3b1b2b3| (1)

Write the expression for dot product between a and b vectors.

ab=a1b1+a2b2+a3b3 (2)

Scalar triple product=a(b×c)

Write the expression to find volume of the parallelepiped (V) .

V=|a(b×c)| (3)

Modify equation (1).

v×w=|ijkv1v2v3w1w2w3|

Substitute 3 for v1 , 1 for v2 , 0 for v3 , 5 for w1 , 9 for w2 and 4 for w3 ,

v×w=|ijk310594|=|109−<

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