Show that the points `A(-1,\ 4,\ -3),\ B(3,\ 2,\ -5),\ C(-3,\ 8,\ -5)a n d\ D(-3,\ 2,\ 1)` are coplanar.

Explanation

$$\text{It is given that}$$ $$A(-1,4,-3), B(3,2,-5), C(-3,8,-5) \text{and}D(-3,2,1)$$
Four points `A, B, C, D` are coplanar if the three vectors
`vec{A B}, vec{A C}` and `vec{A D}` are coplanar.
$$[\overrightarrow{A B}, \overrightarrow{A C}, \overrightarrow{A D}]=0$$
$$\overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}=3 \hat{i}+2 \hat{j}-5 \hat{k}+\hat{1}-4 \hat{j}+3 \hat{k}=4 \hat{1}-2 \hat{j}-2 \hat{k}$$
$$\overrightarrow{A C}=\overrightarrow{O C}-\overrightarrow{O A}=-3 \hat{1}+8 \hat{j}-5 \hat{k}+\hat{1}-4 \hat{j}+3 \hat{k}=-2 \hat{1}+4 \hat{\jmath}-2 \hat{k}$$
$$\overrightarrow{A D}=\overrightarrow{O D}-\overrightarrow{O A}=-3 \hat{i}+2 \hat{j}+\hat{k}+\hat{1}-4 \hat{j}+3 \hat{k}=-2 \hat{1}-2 \hat{j}+4 \hat{k}$$
$$[\overrightarrow{A B}, \overrightarrow{A C}, \overrightarrow{A D}]=\left|\begin{array}{ccc}4 & -2 & -2 \\ -2 & 4 & -2 \\ -2 & -2 & 4\end{array}\right|$$
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