Find the direction cosines of the unit vector perpendicular to the plane ` vec rdot((6 hat i-3 hat j-2 hat k)+1=0` passing through the origin.

Explanation

Given:- `\vec{r} \cdot(6 \hat{i}-3 \hat{j}-2 \hat{k})+1=0`

`\vec{r} \cdot(6 \hat{i}-3 \hat{j}-2 \hat{k})=-1`


`\vec{r} \cdot(-6 \hat{i}+3 \hat{j}+2 \hat{k})=1`


`\vec{n}=-6 \hat{i}+3 \hat{j}+2 \hat{k}`


`\hat{n}=\frac{\vec{n}}{|\vec{n}|}=\frac{-6 \hat{i}+3 \hat{j}+2 \hat{k}}{\sqrt{(-6)^{2}+3^{2}+2^{2}}}`



`=-\frac{6 \hat{i}}{\sqrt{49}}+\frac{3 \hat{j}}{\sqrt{49}}+\frac{2 \hat{k}}{\sqrt{49}}

`=`-\frac{6}{7}\hat{i}+\frac{3}{7} \hat{j}+\frac{2}{7} \hat{k}`


Therefore, direction cosines of the unit vector to the plane `\vec{r}` `=(-\frac{6}{7}+\frac{3}{7}+\frac{2}{7})`