   Chapter 12.4, Problem 6E

Chapter
Section
Textbook Problem

# Find the cross product a × b and verify that it is orthogonal to both a and b.6. a = ti + cos tj + sin tk, b = i − sin tj + cos tk

To determine

To find: The cross product between a and b and verify a×b is orthogonal to both a and b.

Explanation

Given:

a=ti+costj+sintk and b=isintj+costk .

Formula:

Consider the general expression to find the cross product of a and b.

a×b=|ijka1a2a3b1b2b3| (1)

Condition to verify a×b is orthogonal to a.

(a×b)a=0

Condition to verify a×b is orthogonal to b.

(a×b)b=0

In equation (1), substitute t for a1 , cost for a2 , sint for a3 , 1 for b1 , sint for b2 and cost for b3 .

a×b=|ijktcostsint1sintcost|=|costsintsintcost|i|tsint1cost|j+|tcost1sint|k=(cos2t+sin2t)i(tcostsint)j+(tsintcost)k=i+(sinttcost)j+(tsintcost)k

Thus, the a×b is i+(sinttcost)j+(tsintcost)k_ .

Find (a×b)a .

Substitute i+(sinttcost)j+(tsintcost)k for a×b and ti+costj+sintk for a

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