Two adjacent sides of a parallelogram `A B C D` are given by ` vec A B=2 hat i+10 hat j+11 hat ka n d vec A D=- hat i+2 hat j+2 hat kdot` The side `A D` is rotated by an acute angle `alpha` in the plane of the parallelogram so that `A D` becomes `A D^(prime)dot` If `A D '` makes a right angle with the side `A B ,` then the cosine of the angel `alpha` is given by a. `8/9` b. `(sqrt(17))/9` c. `1/9` d. `(4sqrt(5))/9`

Options

A. `(8)/(9)`
B. `(sqrt(17))/(9)`
C. `(1)/(9)`
D. `(4sqrt(5))/(9)`

Answer

B

Explanation

`vec(AB) =2hat(i) + 10hat(j)+11 hat(k), vec(AD) =- hat(j) + 2hat(j) +2hat(k)`
angle 0 between `vec(AB) " and " vec(AD) ` is
`" cos(0) = " |(vec(AB).vec(AD))/(|vec(AB)||vec(AD)|)| = |(-2+20+22)/((15)(3))|=(8)/(9)`
`rArr sin (0) =(sqrt(17))/(9)`
Since `alpha+0 =90^(@)`
`:. cos (alpha) = cos (90^(@)-0) = sin (0) = (sqrt(17))/(9)`