Find a vector of magnitude 9, which is perpendicular to both vectors `4 hat i- hat j+3 hat ka n d-2 hat i+ hat j-2 hat k` .

Options

A. `3hat(i)+6hat(j)-6hat(k)`
B. `3hat(i)-6hat(j)+6hat(k)`
C. `-3hat(i)+6hat(j)+6hat(k)`
D. None of these

Answer

C

Explanation

Let `a =4hati - hatj+3 hatk, b=-2 hati+ hatj-2 hatk`
and `c = x hati + y hatj +z hatk`
Given, `a*c=0`
i.e., ` 4x-y+3z=-0" "…(i)`
and `b*c=0`
i.e, `x^(2)+y^(2)+z^(2)=0" "…(ii)`
Also, `|c|=9`
i.e.,` x^(2)+y^(2)+z^(2)=81" "...(iii)`
Now, from Eqs. (i) and (ii), we get
`2x+z=0impliesz=-2x`
On putting this value in Eq. (ii) by 3 and then adding, we get
`({:(8x-2y+6z=0),(-6x+3y-6z=0):})/(2x+y=0impliesy=-2x)`
On putting this value in Eq. (iv) we get
`5x^(2)+4x^(2)=81`
`=?9x^(2)=81impliesx^(2)=9`
`impliesx = pm 3`
`therefore y= pm 6 and z = pm 6`
`therefore` Required vector, `c=x hati+y hatj +z hatk = pm 3 hatipm 6 hatj pm 6 hatk=3 hati-6 hatj - 6 hatkor -3 hati+6 hatj+6 hatk`