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. `3i + 6j - 6 k `
B. ` 3i - 6j + 6k `
C. `-3i + 6j + 6k `
D. None of the above

Answer

C

Explanation

Let ` a = 4i - j + 3k, b = - 2i + j - 2 k `
and ` c =x i + yj + zk `
Given ` a * c = 0 `
i.e., ` 4 x -y + 3z= 0 " " `… (i)
and ` b * c = 0 `
i.e., ` - 2x + y -2z = 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 = 0 rArr z = - 2x `
On putting this value in Eq. (iii), we get
` x ^ 2 + y ^ 2 + 4 x^ 2 = 81 `
` rArr 5x^ 2 + y ^ 2 = 81 " " `... (iv)
On multiplying Eq. (i) by 2 and Eq. (ii) by 3 and then adding , we get
` {:( 8x - 2y + 6 z= 0 ), (ul(-6x + 3y - 6z = 0 )), ( 2x + y = 0 ) :} `
` rArr y = - 2x `
On putting this value in Eq. (iv) , we get
` 5x ^ 2 + 4x ^ 2 = 8 1 `
` rArr 9 x ^ 2 = 81 `
` rArr x ^ 2 = 9 `
` rArr x = pm 3 `
` therefore y = pm 6 and z = pm 6 `
` therefore ` Required vector, ` c = x i + yj + zk `
` = pm 3i pm 6j pm 6 k `
` = 3i - 6 j - 6k `
` = -3i + 6j + 6k`