Answer
a) $6$
b) $4$
c) $2$
d) $2\sqrt 10$
e) $2\sqrt 13$
f) $2\sqrt 5$
Work Step by Step
a)
All points on the $x$$y$-plane have the form $($$x$, $y$, $0$ $)$, $x$, $y$ $\in$ $R$.
The closest point on the $x$$y$-plane is $($ $4$, $-2$, $0$ $)$,
and the distance is :
$d$ $=$ $ \sqrt {(4-4)^{2}+(-2-(-2))^{2}+(6-0)^{2}}$ $=$ $6$
b)
All points on the $y$$z$-plane have the form $($$0$, $y$, $z$ $)$, $y$, $z$ $\in$ $R$.
The closest point on the $x$$y$-plane is $($ $0$, $-2$, $6$ $)$,
and the distance is :
$d$ $=$ $ \sqrt {(4-0)^{2}+(-2-(-2))^{2}+(6-6)^{2}}$ $=$ $4$
c)
All points on the $x$$z$-plane have the form $($$x$, $0$, $z$ $)$, $x$, $z$ $\in$ $R$.
The closest point on the $x$$y$-plane is $($ $4$, $0$, $6$ $)$,
and the distance is :
$d$ $=$ $ \sqrt {(4-4)^{2}+(-2-0))^{2}+(6-6)^{2}}$ $=$ $2$
d) All points on the $x$-$axis$ have the form $($$x$, $0$, $0$ $)$, $x$ $\in$ $R$.
The closest point is $($ $4$, $0$, $0$ $)$,and the distance is :
$d$ $=$ $ \sqrt {(4-4)^{2}+(-2-0))^{2}+(6-0)^{2}}$ $=$ $2\sqrt 10$
e)All points on the $y$-$axis$ have the form $($$0$, $y$, $0$ $)$, $y$ $\in$ $R$.
The closest point is $($ $0$, $-2$, $0$ $)$,and the distance is :
$d$ $=$ $ \sqrt {(4-0)^{2}+(-2-(-2))^{2}+(6-0)^{2}}$ $=$ $2\sqrt 13$
g) All points on the $z$-$axis$ have the form $($$0$, $0$, $z$ $)$, $z$ $\in$ $R$.
The closest point is $($ $0$, $0$, $6$ $)$,and the distance is :
$d$ $=$ $ \sqrt {(4-0)^{2}+(-2-0)^{2}+(6-6)^{2}}$ $=$ $2\sqrt 5$
