What is the interior acute angle of the parallelogram whose sides are represented by the vectors `(1)/(sqrt(2))hat(i)+(1)/(sqrt(2))hat(j)+hat(k) and (1)/(sqrt(2))hat(i) - (1)/(sqrt(2))hat(j)+hat(k)` ?

Options

A. `60^(@)`
B. `45^(@)`
C. `30^(@)`
D. `15^(@)`

Answer

A

Explanation

Let `vec(a)=(1)/(sqrt(2))hat(i)+(1)/(sqrt(2))hat(j)+hat(k) `
`vec(b)=(1)/(sqrt(2))hat(i)-(1)/(sqrt(2))hat(j)+hat(k)`
`:.cos theta=(vec(a).vec(b))/(|vec(a)||vec(b)|)`
`=(((1)/(sqrt(2))hat(i)+(1)/(sqrt(2))hat(j)+hat(k)).((1)/(sqrt(2))hat(i)-(1)/(sqrt(2))hat(j)+hat(k)))/(sqrt(1/2+1/2+1)sqrt(1/2+1/2+1))`
`=(1)/(2)[(1)/(2)-(1)/(2)+1]=1/1=cos60^(@)`
`:. theta 60^(@)`