`bar(OA)=bar(i)+2bar(j)+2bar(k)` .In the plane `bar(OA)` and `bar(i)` rotate `bar(OA)` through `90^(@)` about the origin `o` such that the new position of `bar(OA)` can

Options

A. `(1)/(sqrt(2))(4hat(i)-hat(j)-hat(k))`
B. `(1)/(sqrt(2))(-4hat(i)+hat(j)+hat(k))`
C. `sqrt(2)(2hat(i)-2hat(k))`
D. `sqrt(2)(6hat(i)-3hat(k))`

Answer

A

Explanation

Given `vec(OA)=hat(i)+2hat(j)+2hat(k)`
Let new position of `vec(OA) " is " vec(r)=ahat(i)+bhat(j)+chat(k)`
`because vec(OA),vec(r) " and " hat(i) " are coplaner" =|(1,2,2),(a,b,c),(1,0,0)| = 0 implies b = c`
`because vec(r)_|_vec(OA) implies a+2b+2c=0impliesa=-4b{because b=c}`
`therefore vec(r)=-4bhat(j)+bhat(j)+bhat(k)=-b(4hat(i)-hat(j)-hat(k))`
Also `|vec(r)|=|vec(OA)|impliesb=pm(1)/(sqrt(2))impliesvec(r)=pm(1)/(sqrt(2))(4hat(i)-hat(j)-hat(k))`
`because vec(r)` makes acute angle with positive x-axis `implies vec(r)=(1)/(sqrt(2))(4hat(i)-hat(j)-hat(k))`