Question

If the four points with position vectors $-2\hat{i}+\hat{j}+k, \hat{i} +\hat{j}+\hat{k}, \hat{j}-\hat{k}, $ and $\lambda\hat{j}+\hat{k}$ are coplanar, then $\lambda=$

• A. 1
• B. 2
• C. -1
• D. 0

Answer 1

Answer: (A)

If four points $\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right),\left(x_{3}, y_{3}, z_{3}\right)$ and
$\left(x_{4}, y_{4}, z_{4}\right)$ are coplanar, then
$\begin{vmatrix}x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\ x_{3}-x_{1} & y_{3}-y_{1} & z_{3}-z_{1} \\ x_{4}-x_{1} & y_{4}-y_{1} & z_{4}-z_{1}\end{vmatrix}=0$
Now, $\begin{vmatrix}3 & 0 & 0 \\ 2 & 0 & -2 \\ 2 & \lambda-1 & 0\end{vmatrix}=0$
$\Rightarrow 3(0+2 \lambda-2)=0$
$\Rightarrow \lambda=1$