Question

If four points A, B, C and D with position vectors 4$\hat{i}+$3$\hat{j}+$3$\hat{k}$, 5$\hat{i}+$ $x\hat{j}+$ 7$\hat{k}$, 5$\hat{i}+$3$\hat{j}$ and $7\hat{i}+6\hat{j}+\hat{k}$ respectively are coplanar, then find the value of x.

Answer 1

Let $\overrightarrow{\mathrm{OA}}=4\hat{i}+3\hat{j}+3\hat{k},\overrightarrow{\mathrm{OB}}=5\hat{i}+x\hat{j}+7\hat{k},\overrightarrow{\mathrm{OC}}=5\hat{i}+3\hat{j}$ and $\overrightarrow{\mathrm{OD}}=7\hat{i}+6\hat{j}+\hat{k}$.

$\therefore \overrightarrow{\mathrm{AB}}=\left(5\hat{i}+x\hat{j}+7\hat{k}\right)-\left(4\hat{i}+3\hat{j}+3\hat{k}\right)=\hat{i}+\left(x-3\right)\hat{j}+4\hat{k}$

$\overrightarrow{\mathrm{AC}}=\left(5\hat{i}+3\hat{j}\right)-\left(4\hat{i}+3\hat{j}+3\hat{k}\right)=\hat{i}-3\hat{k}$

$\overrightarrow{\mathrm{AD}}=\left(7\hat{i}+6\hat{j}+\hat{k}\right)-\left(4\hat{i}+3\hat{j}+3\hat{k}\right)=3\hat{i}+3\hat{j}-2\hat{k}$

Since the given four points are coplanar, so the vectors $\overrightarrow{\mathrm{AB}},\overrightarrow{\mathrm{AC}}$ and $\overrightarrow{\mathrm{AD}}$ are also coplanar.

$\therefore \left[\begin{array}{ccc}\overrightarrow{\mathrm{AB}}& \overrightarrow{\mathrm{AC}}& \overrightarrow{\mathrm{AD}}\end{array}\right]=0$

$$\begin{gathered} \Rightarrow \left|\begin{array}{ccc}1& x-3& 4\\ 1& 0& -3\\ 3& 3& -2\end{array}\right|=0 \\ \Rightarrow 1\left(0+9\right)-\left(x-3\right)\left(-2+9\right)+4\left(3-0\right)=0 \\ \Rightarrow 9-7x+21+12=0 \\ \Rightarrow 7x=42 \\ \Rightarrow x=6 \end{gathered}$$
Thus, the value of x is 6.