Question

If 0 is the origin and A is (a,b,c), then find the direction cosines of the line OA and the equation of plane through A at right angle OA.

Answer 1

Since, DC's of line OA are $\frac{a}{\sqrt{a^2+b^2+c^2}},\frac{b}{\sqrt{a^2+b^2+c^2}}and\frac{c}{\sqrt{a^2+b^2+c^2}},$
Also, $\overrightarrow{n}=\overrightarrow{OA}=\overrightarrow{a}=\hat{i}+b\hat{j}+c\hat{k}$

The equation of plane passes through (a,b,c) and perpendicular to OA is given by $[\overrightarrow{r}-\overrightarrow{a}].\overrightarrow{n}=0.\Rightarrow \overrightarrow{r}.\overrightarrow{n}=\overrightarrow{a}.\overrightarrow{n}\Rightarrow \left[\right(x\hat{i}+y\hat{j}+z\hat{k}).(a\hat{i}+b\hat{j}+c\hat{k}\left)\right]=(a\hat{i}+b\hat{j}+c\hat{k})\Rightarrow ax+by+cz=a^2+b^2+c^2$