Determine the relationship between the coordinates of point M (x,y,z) so that the points M, A, and B are collinear

Question

So the two points A (0,0,5) and B (1,1,0) were given, and since all three points have to be collinear, I did the following:

$\overrightarrow(MA)=k\overrightarrow(AB)$

$\begin{pmatrix}0-x\\\ 0-y \\\ 5-z\end{pmatrix}=k\begin{pmatrix}1-0\\\ 1-0 \\\ 0-5\end{pmatrix}$

$\Rightarrow -x=k$ $\Rightarrow x=-k$

$\Rightarrow -y=k$ $\Rightarrow y=-k$

$\Rightarrow 5-z=-5k$ $\Rightarrow z=5+5k$

But, the answer is $x=k$, $y=k$, $z=5-5k$.

What did I do wrong ?

Answer 1

You are correct.

Let $h=-k$

$(-k,-k,5+5k)=(h,h,5-5h)$