What is the value of m if the vectors
`2hat(i)-hat(j)+hat(k), hat(i)+2hat(j)-3hat(k) and 3hat(i)+m hat(j)+5 hat(k)` are coplanar?

Options

A. -2
B. 2
C. -4
D. 4

Answer

C

Explanation

Three vectors `x_(1)vec(a)+y_(1)vec(b)+z_(1)vec(c), x_(2)vec(a)+y_(2)vec(b)+z_(2)vec(c) and x_(3)vec(a)+y_(3)vec(b)+z_(3) vec(c)` will be coplanar iff
`|(x_(1),y_(1),z_(1)), (x_(2), y_(2), z_(2)),(x_(3), y_(3), z_(3))|=0`
Here, `x_(1)=2, y_(1)=-1, z_(1)=1`
`x_(2)=1, y_(2)=2, z_(2)=-3`
`x_(3)=3, y_(3)=m, z_(3)=5`
`:. |(2,-1,1),(1,2,-3),(3, m, 5)|=0`
`rArr 2(10+3m)+1(5+9)+1(m-6)=0`
`rArr 7m +28=0 rArr m=-4`