Question

Find $\lambda$ if the vectors $\hat{i}-\hat{j}+\hat{k},3\hat{i}+\hat{j}+2\hat{k},\hat{i}+\lambda \hat{j}-3\hat{k}$ are coplanar

Answer 1

$a=i-j+k$, $b=3i+j+2k$, $c=i+\lambda j-3k$

If $a,b,c$ are coplanar then

$\left[abc\right]=0$

i.e., $(a\times b).c=0$

or $(c\times a).b=0$

Now let us use determinant concept

$\left|\begin{array}{ccc}1& -1& 1\\ 3& 1& 2\\ 1& \lambda & -3\end{array}\right|=0$

$1(-3-2\lambda )+1(-9-2)+1(3\lambda -)=0$

$-3-2\lambda -9-2+3\lambda -1=0$

$\lambda -15=0$

$\lambda =15$