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 `8/9` b. `(sqrt(17))/9` c. `1/9` d. `(4sqrt(5))/9`

Options

A. `8/9`
B. `sqrt17/9`
C. `1/9`
D. `(4sqrt5)/9`

Answer

b

Explanation

Angle between vectors `vecAB and vecAD ` is given by
`cos theta= (vecAB.vecAD)/(|vecAB|.|vecAD|)= (-2+20+22)/(sqrt(4+100+121)sqrt(1+4+4))`
`= 8/9`
`Rightarrow cos alpha = cos( 90^(@) -theta) = sin theta= sqrt17/9`