Question

Let $\overrightarrow{ x }$ be a vector in the plane containing vectors $\overrightarrow{ a }=2 \hat{ i }-\hat{ j }+\hat{ k }$ and $\overrightarrow{ b }=\hat{ i }+2 \hat{ j }-\hat{ k }$. If the vector $\overrightarrow{ x }$ is perpendicular to $(3 \hat{ i }+2 \hat{ j }-\hat{ k })$ and its projection on $\vec{a}$ is $\frac{17 \sqrt{6}}{2}$, then the value of $|\overrightarrow{ x }|^{2}$ is equal to ________.

Answer 1

Let $\overrightarrow{ x }=\lambda \overrightarrow{ a }+\mu \overrightarrow{ b } (\lambda$ and $\mu$ are scalars $)$
$\overrightarrow{ x }=\hat{ i }(2 \lambda+\mu)+\hat{ j }(2 \mu-\lambda)+\hat{ k }(\lambda-\mu)$
Since $\overrightarrow{ x } \cdot(3 \hat{ i }+2 \hat{ j }-\hat{ k })=0$
$3 \lambda+8 \mu=0 \ldots \ldots(1)$
Also Projection of $\overrightarrow{ x }$ on $\overrightarrow{ a }$ is $\frac{17 \sqrt{6}}{2}$
$\frac{\overrightarrow{ x } \cdot \overrightarrow{ a }}{|\overrightarrow{ a }|}=\frac{17 \sqrt{6}}{2}$
$6 \lambda-\mu=51 \dots$(2)
From (1) and (2)
$\lambda=8, \mu=-3$
$\overrightarrow{ x }=13 \hat{ i }-14 \hat{ j }+11 \hat{ k }$
$|\overrightarrow{ x }|^{2}=486$