If ` veca , vecb` are unit vectors such that the vector ` veca + 3vecb ` is peependicular to ` 7 veca - vecb and veca -4vecb` is prependicular to ` 7 veca -2vecb` then the angle between ` veca and vecb` is

Options

A. ` pi//6`
B. ` pi//4`
C. ` pi//3`
D. ` pi//2`

Answer

C

Explanation

Let ` theta` be the angle between `veca and vecb` is
we have,
` ( veca + 3vecb) bot ( 7 veca - 5vecb)`
` Rightarrow ( veca + 3vecb) . ( 7 veca -5vecb) =0 `
` Rightarrow 7|veca|^(2) + 16(veca .vecb) -15 |vecb|^(2) =0`
` Rightarrow 15-30 cos theta =0 Rightarrow cos theta = 1/2 Rightarrow 1/2 Rightarrow theta = pi/3`
And,
` (veca - 4 vecb ) bot ( 7veca - 2vecb)`
` Rightarrow (veca -4vecb) . ( 7 veca - 2 vecb) =0`
` Rightarrow 7| veca|^(2) + 8|vecb|^(2) -30 (veca .vecb) =0`
` Rightarrow 15-30 cos theta =0 Rightarrow = 1/2 Rightarrow = pi /3`