Question

Let $\overrightarrow{ a }=2 \hat{ i }+\hat{ j }-2 \hat{ k }$ and $\overrightarrow{ b }=\hat{ i }+\hat{ j }$. If $\overrightarrow{ c }$ is a vector such that $\overrightarrow{ a } \cdot \overrightarrow{ c }=|\overrightarrow{ c }|,|\overrightarrow{ c }-\overrightarrow{ a }|=2 \sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $30^{\circ}$, then find value of $|(\vec{a} \times \vec{b}) \times \vec{c}|$.

Answer 1

$\vec{ a } \times \vec{ b }=2 \hat{ i }-2 \hat{ j }+\hat{ k }$
$|\vec{ a } \times \vec{ b }|=3$
$|\vec{ c }-\vec{ a }|=2 \sqrt{2}$
$|\vec{ c }-\vec{ a }|^{2}=8$
$|\vec{ c }|^{2}+|\vec{ a }|^{2}-2 \vec{ a } \cdot \vec{ c }=8$
$|\vec{ c }|^{2}+|\vec{ a }|^{2}-2 \vec{ a } \cdot \vec{ c }=8$
$|\vec{ c }|^{2}+9-2|\vec{ c }|=8$
$\therefore(|\vec{ c }|-1)^{2}=0$
$|\vec{ c }|=1$
$\therefore|(\vec{ a } \times \vec{ b }) \times \vec{ c }|$
$=|\vec{ a } \times \vec{ b }||\vec{ c }| \sin 30^{\circ}=\frac{3}{2}$