Question

Let a = i - j, b = j - k, c = k - i. Find the unit vector d such that a.d = 0 = [b c d]

Answer 1

Let us consider d = x i + y j + z k

then, a.d = (i - j).(x i + y j + z k)

a.d = x - y = 0

x = y .......(1)

[b c d] = 0

b.(c × d) = 0

b.[(k - i) × (x i + yj + zk)] = 0

(j - k).[xj -y i -yk + zj ] = 0

(j - k).{(x + z)j - yi - yk} = 0

x + z + y = 0

x + y + z = 0

x + x + z = 0 [ from equation (1) ]

z = -2x

now, d = xi + xj - 2xk

d is unit vector , so |d| = 1

√{x² + x² + 4x²} = 1

6x² = 1

x = ±1/√6

hence, d = ±1/√6(i + j - 2k)