Mathematics

Problem

Let

a, b

and

c

are three vectors having magnitudes 1, 2 and 3, respectively satisfy the relation

[a b c] = 6

. If

\hat{d}

is a unit vector coplanar with

b

and

c

such that

b \cdot \hat{d} = 1

, then evaluate

[(a \times c) \cdot \hat{d}]^2 + [(a \times c) \times \hat{d}]^2