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Question

Let a=^i+4^j+2^k,b=3^i2^j+7^k and c=2^i^j+4^k. Find a vector p which is perpendicular to both a and b and pc=18.

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Solution

a=^i+4^j+2^k,b=3^i2^j+7^k
Since we need a vector perpendicular to both a and b, it has to be parallel to a×b
a×b=(^i+4^j+2^k)×(3^i2^j+7^k)
=2^k7^j12^k+28^i+6^j+4^i
=32^i^j14^k
The unit vector in this direction can be written as 32^i^j14^k1024+1+196=32^i^j14^k1221
Since p.c=18,
k1221×(32^i^j14^k).(2^i^j+4^k)=18
k1221×(64+156)=18
k=21221
p=k1221×(32^i^j14^k)=2×(32^i^j14^k)=64^i2^j28^k

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