compute parameter m if vectors are co-planer

Question

What is the value of m in vector c so that a, b, and c are on the same plane?

$a=(2, -2, -1)$
$b=(-3, 2, 1)$
$c=(m, 2, m)$

I think if vectors are coplanar they should lay in one plane => (a x b) * c = 0, so 2*2*m -2*m -6 -2*m -6*m - 4 = 0 . And i got m= -5/3 , am i correct?

Answer 1

You want to have the determinant $$ \det\begin {bmatrix}2&-2&-1\\-3&2&1\\m&2&m\end {bmatrix} =0$$

I found $m=1$ and checked it.

Answer 2

As an alternative, let $ax+by=c$ we obtain

• $2x-3y=m $
• $-2x+2y=2$
• $-x+y=m$

the associated augmented matrix is

$$\begin {vmatrix}2&-3&m\\-2&2&2\\-1&1&m\end {vmatrix}\to \begin {vmatrix}2&-3&m\\-2&2&2\\-2&2&2m\end {vmatrix}\to \begin {vmatrix}2&-3&m\\-2&2&2\\0&0&2m-2\end {vmatrix}$$

thus the system has a unique solution $\iff m=1$ that is

• $2x-3y=1 $
• $-2x+2y=2$

from which we obtain $x=-4,y=-3$ indeed

$$-4\cdot (2,-2,-1)-3\cdot (-3,2,1)=(1,2,1)$$