Question:
if p,q,r denote the vectors b*c , c*a and a*b respectively , show that a is parallel to q*r , b to r*p , c to p*q respectively
Answer:
We have a * (q * r) = (a . r)q - (a . q)r
= {a . (a * b)}q - {a. (c * a)}r
= [a a b]q - [a c a]r
= 0 * q - 0 * r {Since [a a b] = [a c a] = 0}
= 0
Since, a * (q * r) = 0
Hence, a is parallel to q * r
We have b * (r * p) = (b . p)r - (b . r)p
= {b . (b * c)}r - {b. (a * b)}p
= [b b c]r - [b a b]p
= 0 * r - 0 * p {Since [b b c] = [b a b] = 0}
= 0
Since, b * (r * p) = 0
Hence, b is parallel to r * p
We have c * (p * q) = (c . p)q - (c . q)p
= {c . (b * c)}q - {c. (c * a)}p
= [c b c]q - [c c a]p
= 0 * q - 0 * p {Since [c b c] = [c c a] = 0}
= 0
Since, c * (p * q) = 0
Hence, c is parallel to p * q
Hence, a is parallel to q * r , b to r * p , c to p * q respectively.
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