Question

The value of k for which the points (1,0,3) (-1,3,4) (1,2,1) (k,2,5) are coplanar is

Answer 1

Answer:

The value of k is -1

Step-by-step explanation:

Let the given points be

A(1,0,3), B(-1,3,4), C(1,2,1), D(k,2,5)

Then,

$\overrightarrow{OA}=\overrightarrow{i}-0\overrightarrow{j}+3\overrightarrow{k}$

$\overrightarrow{OB}=-\overrightarrow{i}+3\overrightarrow{j}+4\overrightarrow{k}$

$\overrightarrow{OC}=\overrightarrow{i}+2\overrightarrow{j}+\overrightarrow{k}$

$\overrightarrow{OD}=k\overrightarrow{i}+2\overrightarrow{j}+5\overrightarrow{k}$

$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=-2\overrightarrow{i}+3\overrightarrow{j}+\overrightarrow{k}$

$\overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=0\overrightarrow{i}+2\overrightarrow{j}-2\overrightarrow{k}$

$\overrightarrow{AD}=\overrightarrow{OD}-\overrightarrow{OA}=(k-1)\overrightarrow{i}+2\overrightarrow{j}+2\overrightarrow{k}$

$\text{ since }\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AD}\text{ are coplanar, }[\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AD}]=0$

$\implies\left|\begin{array}{ccc}-2&3&1\\0&2&-2\\(k-1)&2&2\end{array}\right|=0$

$\implies\,-2(4+4)-3(0+2k-2)+1(-2k+2)=0$

$\implies\,-16-6k+6-2k+2=0$

$\implies\,-8-8k=0$

$\implies\,-8k=8$

$\implies\,\boxed{\bf\,k=-1}$