Let vector a = -i -k, b = -i+j and c = i+2j+3k be three given vectors. If vector r is a vector such that r x b =c.b and r.a =0, then the value of r.b is ...............

(r, a, b, c are the vactor quantity)

Ans. 9

Let r = xi + yj + zk

According to 1st condition,

r*b = c.b

or (xi + yj + zk)*(-i + j) = (i + 2j + 3k).(-i + j)

or (x)(1) + (x)(0) + (y)(-1) + (y)(0) + (z)(-1) +(z)(1) = -1 + 2 + 0

or x - y = 1

or y = x - 1------------------(1)

According to 2nd condition,

r.a = 0

or (xi + yj + zk)(-i - k ) = 0

or -x - z + 0 = 0

or x+z = 0

or z = -x-------------(2)

Now, r.b = (xi + yj + zk)(-i + j)

or r.b = [xi +(x - 1)j -xk](-i + j) [From (1) and (2)]

or r.b = (-x) + (x - 1) + 0

or r.b = -1

Hence, r.b = -1.