If the vector $\overline{\mathrm{c}}$ lies in the plane of $\overline{\mathrm{a}}$ and $\bar{b}$, where…
If the vector $\overline{\mathrm{c}}$ lies in the plane of $\overline{\mathrm{a}}$ and $\bar{b}$, where $\bar{a}=\hat{i}-\hat{j}+2 \hat{k}, \bar{b}=\hat{i}+\hat{j}+\hat{k}$ and $\overline{\mathrm{c}}=x \hat{\mathrm{i}}-(2-x) \hat{\mathrm{j}}-\hat{\mathrm{k}}$, then the value of $x$ is
4
-4
2
-2
Solution
According to the given condition, vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are coplanar.
$\begin{aligned}
& \therefore \quad \overline{\mathrm{a}} \cdot(\overline{\mathrm{~b}} \times \overline{\mathrm{c}})=0 \\
& \therefore \quad\left|\begin{array}{ccc}
1 & -1 & 2 \\
1 & 1 & 1 \\
x & x-2 & -1
\end{array}\right|=0 \\
& \Rightarrow 1(-1-x+2)+1(-1-x)+2(x-2-x)=0 \\
& \\
& \Rightarrow 1-x-1-x-4=0 \\
& \Rightarrow x=-2
\end{aligned}$