Question

State true or false:

The points $A(4,5,1);B(5,3,4);C(4,1,6)$ and $D(3,3,3)$ are coplanar.

• A. True
• B. False

Answer 1

The correct option is A True

$A(4,5,1),B(5,3,4);C(4,1,6)$ & $D(3,3,3)$

$\overrightarrow{AB}=(5-4)\hat{i}+(3-5)\hat{j}+(4-1)\hat{k}$

$=\hat{i}-2\hat{j}+3\hat{k}$

$\overrightarrow{AC}=(4-4)\hat{i}+(1-5)\hat{j}+(6-1)\hat{k}$

$=-4\hat{j}+5\hat{k}$

$\overrightarrow{AD}=(3-4)\hat{i}+(3-5)\hat{j}+(3-1)\hat{k}$

$=-\hat{i}-2\hat{j}+2\hat{k}$

$\therefore |\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AD}|=\left|\begin{array}{ccc}1& -2& 3\\ 0& -4& 5\\ -1& -2& 2\end{array}\right|$

$=1(-8+10)+2\left(\right(2\times 0)+5)+3\left(\right(-2\times 0)-4)$

$=2+10-12=0$

Hence four points A, B, C, D afe coplanar.