Question

If $\vec{a}, \vec{b}$ are vectors perpendicular to each other and $|\vec{a}|=2,|\vec{b}|=3, \vec{c} \times \vec{a}=\vec{b}$, then the least value of $|\vec{c}-\vec{a}|$ is ______ .

Answer 1

$ \vec{c} \times \vec{a} =\vec{b}$
$ \Rightarrow |\vec{c} \times \vec{a}|=|\vec{b}|$
$ \Rightarrow |\vec{c}||\vec{a}| \sin \theta=3$
$ \Rightarrow |\vec{c}|=\frac{3}{2 \sin \theta} $
$|\vec{c}-\vec{a}|^{2} =|\vec{c}|^{2}+|\vec{a}|^{2}-2 \vec{c} \cdot \vec{a}$
$=|\vec{c}|^{2}+4-2|\vec{c}| \cdot|\vec{a}| \cos \theta $
$=\frac{9}{4 \sin ^{2} \theta}+4-2 \cdot \frac{3}{2 \sin \theta} \cdot 2 \cdot \cos \theta$
$=4+\frac{9}{4} \operatorname{cosec}^{2} \theta-6 \cot \theta$
$=\frac{9}{4}+\left(\frac{3}{2} \cot \theta-2\right)^{2}$
$\Rightarrow |\vec{c}-\vec{a}|^{2} \geq \frac{9}{4}$
$ \Rightarrow|\vec{c}-\vec{a}| \geq \frac{3}{2}$