The sum of the distinct real values of $\mu$, for which the vector $\mu \hat{i}+\hat{j}+\hat{k}, \hat{i}+\mu…
The sum of the distinct real values of $\mu$, for which the vector $\mu \hat{i}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}$ are coplanar is
1
-1
2
0
Solution
From the condition for co-planarity
$\begin{aligned}
& \left|\begin{array}{lll}
\mu & 1 & 1 \\
1 & \mu & 1 \\
1 & 1 & \mu
\end{array}\right|=0 \\
& \Rightarrow(2+\mu)(\mu-1)^2=0 \\
& \Rightarrow \mu=-2 \text { or } \mu=1 \\
& \text { sum }=-2+1=-1
\end{aligned}$