Q. Show that the points whose position vectors are $4\hat{i}+5\hat{k}, \hat{i}+\hat{j}+3\hat{k} and -5\hat{j}+3\hat{j}-\hat{k} are collinear.$

Dear Student,
Here is the solution of your asked query:

Suppose A, B and C are the points which are represented by;
$\overrightarrow{\mathrm{A}}= 4i+5k ; \overrightarrow{\mathrm{B}} = \mathrm{i}+\mathrm{j}+3\mathrm{k} \mathrm{and} \overrightarrow{\mathrm{C}}= -5\mathrm{i}+3\mathrm{j}-\mathrm{k}$
Coordinates of these points are A(4, 0, 5); B(1, 1, 3) and C(-5, 3, -1).
So, $$\begin{gathered} \overrightarrow{\mathrm{AB} }= \left(1-4,1-0,3-5\right) = \left(-3,1,-2\right) \\ \overrightarrow{\mathrm{AC} } = \left(-5-4,3-0,-1-5\right) = \left(-9,3,-6\right) \end{gathered}$$
$\left|\begin{array}{ccc}-3& 1& -2\\ -9& 3& -6\end{array}\right| = -3\left(-6+6\right)+1\left(18-18\right)-2\left(18-18\right) = 0$
Regards
Therefore, points A, B and C are collinear.