Question

The sum of the distinct real values of $\mu ,$ for which the vectors, $\mu \hat{i}+\hat{j}+\hat{k},\hat{i}+\mu \hat{j}+\hat{k},\hat{i}+\hat{j}+\mu \hat{k}$ are co-planar, is:

• A. $0$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

The correct option is B $-1$

If the three vecors are coplanar, then $\left|\begin{array}{ccc}\mu & 1& 1\\ 1& \mu & 1\\ 1& 1& \mu \end{array}\right|=0$
$\Rightarrow \mu ({\mu }^2-1)-1(\mu -1)+1(1-\mu )=0$
$\Rightarrow {\mu }^3-\mu -\mu +1+1-\mu =0$
$\Rightarrow {\mu }^3-3\mu +2=0$
$\Rightarrow (\mu -1)({\mu }^2+\mu -2)=0$
$\Rightarrow (\mu -1)(\mu -1)(\mu +2)=0$
$\Rightarrow \mu =1,1,-2$
So, distinct real values of $\mu$ are $1$ and $-2$.
Therefore, thier sum $=-1$