Question

$\overrightarrow{n}$is a vector of magnitude $\sqrt{3}$ and is equally inclined to an acute angle with the coordinate axes. Find the vector and Cartesian forms of the equation of a plane which passes through (2, 1, −1) and is normal to $\overrightarrow{n}$.

Answer 1

$$\begin{gathered} \text{Let α, β and γ be the angles made by}\overrightarrow{n}\text{with}x,y\text{and}\mathit{\text{z}}\text{-axes respectively.} \\ \text{Given that} \\ \alpha =\beta =\gamma \\ \Rightarrow \cos \alpha =\cos \beta =\cos \gamma \\ \Rightarrow l=m=n,\text{where}l,m,n\text{are direction cosines of}\overrightarrow{n}\text{.} \\ \text{But}{l}^2+m^2+n^2=1 \\ \Rightarrow l^2+l^2+l^2=1 \\ \Rightarrow 3l^2=1 \\ \Rightarrow l^2=\frac{1}{3} \\ \Rightarrow l=\frac{1}{\sqrt{3}}\text{(Since α is acute,}\mathit{\text{l}}\text{= cos α >0)} \\ \text{Thus,}\overrightarrow{n}=\sqrt{3}\left(\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}\right)=\hat{i}+\hat{j}+\hat{k}\text{(Using}\overrightarrow{r}\text{=}\left|\overrightarrow{r}\left(l\hat{i}+m\hat{j}+n\hat{k}\right)\right|\text{)} \\ \text{We know that the vector equation of the plane passing through a point}\overrightarrow{a}\text{and normal to}\overrightarrow{n}\text{is} \\ \overrightarrow{r}.\overrightarrow{n}=\overrightarrow{a}.\overrightarrow{n} \\ \text{Substituting}\overrightarrow{a}\text{= 2}\hat{i}\text{+}\hat{j}\text{-}\hat{k}\text{and}\overrightarrow{n}\text{=}\hat{i}+\hat{j}+\hat{k}\text{, we get} \\ \overrightarrow{r}.\left(\hat{i}+\hat{j}+\hat{k}\right)=\left(2\hat{i}\text{+}\hat{j}\text{-}\hat{k}\right).\left(\hat{i}+\hat{j}+\hat{k}\right) \\ \Rightarrow \overrightarrow{r}.\left(\hat{i}+\hat{j}+\hat{k}\right)=2+1-1 \\ \Rightarrow \overrightarrow{r}.\left(\hat{i}+\hat{j}+\hat{k}\right)=2 \\ \text{For the Cartesian form, we need to substitute}\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}\text{in the vector equation.} \\ \text{Then, we get} \\ \left(x\hat{i}+y\hat{j}+z\hat{k}\right).\left(\hat{i}+\hat{j}+\hat{k}\right)=2 \\ \Rightarrow x+y+z=2 \end{gathered}$$