SOLUTION: Determine two vectors that are perpendicular to each other and also perpendicular to u = [4, -3, 1].

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Question 23639: Determine two vectors that are perpendicular to each other and also perpendicular to u = [4, -3, 1].
Answer by venugopalramana(3286) About Me  (Show Source):
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TRUST YOU KNOW VECTOR MULTIPLICATION ...DOT AND CROSS PRODUCTS...OTHERWISE PLEASE INFORM YOUR PRESENT COURSE OF STUDY AND YOUR BACKGROUND KNOWLEDGE TO ENABLE ME GIVE YOU ALTERNATE SOLUTIONS.
Determine two vectors that are perpendicular to each other and also perpendicular to u = [4, -3, 1].THAT IS U=4i-3j+k
LET ONE VECTOR PERPENDICULAR TO U BE T
LET T=Xi+Yj+Zk
SINCE T IS PERPENDICULAR TO U...T.U=0
(Xi+Yj+Zk).(4i-3j+k)=0
4X-3Y+Z=0....HENCE ANY SET OF VALUES OF X,Y AND Z SATISFYING THIS EQUATION WILL MAKE T PERPENDICULAR TO U.
LET US TAKE ARBITRARILY
X=1 , Y=1 AND Z=-1....HENCE T=i+j-k
NOW WE SHOULD FIND ANOTHER VECTOR SAY V WHICH IS PERPENDICULAR TO BOTH T and U.
HENCE V IS PARALLEL TO TXU
TXU=(i+j-k)X(4i-3j+k)= -3k-j-4k+i-4j-3i=-2i-5j-7k
HENCE ANY VECTOR PARALLEL TO -(2i+5j+7k) WILL BE PERPENDICULAR TO BOTH T AND U ..SO
V=(2i+5j+7k)
HENCE T=i+j-k AND V=(2i+5j+7k) ARE 2 VECTORS PERPENDICULAR TO U=4i-3j+k