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

Answer: 1.5

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