Question

Find $p$ if the vectors $a+b+c,a+pb+2c,-a+b+c$ are coplanar where $a,b,c$ are non coplanar.

Answer 1

Since $\overline{a}+\overline{b}+\overline{c},\overline{a}+\overline{pb}+\overline{2c}$ & $\overline{-a}+\overline{b}+\overline{c}$ are coplanar.

$\left|\begin{array}{ccc}1& 1& 1\\ 1& p& 2\\ -1& 1& 1\end{array}\right|=0$

$1(p-2)-1(1+2)+1(1+p)=0$

$p-2-3+1+p=0$

$2p-4=0\Rightarrow p=\frac{4}{2}$

$p=2$