Question Number 31264 by ajfour last updated on 04/Mar/18
$${Find}\:{the}\:{shortest}\:{distance}\:{from}\:{the} \\ $$$${plane}\:\bar {\boldsymbol{{r}}}.\bar {\boldsymbol{{n}}}=\boldsymbol{{q}}\:\:{to}\:{the}\:{sphere} \\ $$$$\:\mid\bar {\boldsymbol{{r}}}−\bar {\boldsymbol{{r}}}_{\mathrm{0}} \mid=\boldsymbol{{R}}\:\:. \\ $$
Commented by ajfour last updated on 04/Mar/18
Commented by ajfour last updated on 05/Mar/18
$${let}\:{the}\:{shortest}\:{distance}\:{be}\:{d}_{{s}} . \\ $$$$\:\mid\left(\bar {{r}}_{\mathrm{0}} −\bar {{r}}_{{A}} \right).\hat {{n}}\:\mid={R}+{d}_{{s}} \\ $$$$\Rightarrow\:\:\:\boldsymbol{{d}}_{\boldsymbol{{s}}} =\frac{\mid\bar {\boldsymbol{{r}}}_{\mathrm{0}} .\bar {\boldsymbol{{n}}}−\boldsymbol{{q}}\mid}{\mid\bar {\boldsymbol{{n}}}\mid}−\boldsymbol{{R}}\:\:. \\ $$