Question

Determine the value of λ for which the following planes are perpendicular to each other.
(i) $\overrightarrow{r}·\left(\hat{i}+2\hat{j}+3\hat{k}\right)=7\mathrm{and}\overrightarrow{r}·\left(\lambda \hat{i}+2\hat{j}-7\hat{k}\right)=26$

(ii) 2x − 4y + 3z = 5 and x + 2y + λz = 5

(iii) 3x − 6y − 2z = 7 and 2x + y − λz = 5

Answer 1

$$\begin{gathered} \left(i\right)\text{We know that the planes}\overrightarrow{r}.\overrightarrow{n_1}=d_{\mathit{1}}\mathit{,}\mathit{}\overrightarrow{r}\mathit{.}\mathit{}\overrightarrow{n_{\mathit{2}}}=d_2\text{are perpendicular to each other only if}\overrightarrow{n_1}\text{.}\overrightarrow{n_2}\text{=0.} \\ \text{Here,}\overrightarrow{n_1}=\hat{i}+2\hat{j}+3\hat{k};\overrightarrow{n_2}=\lambda \hat{i}+2\hat{j}-7\hat{k} \\ \text{The given planes are perpendicular.} \\ \Rightarrow \overrightarrow{n_1}.\overrightarrow{n_2}=0 \\ \Rightarrow \left(\hat{i}+2\hat{j}+3\hat{k}\right).\left(\lambda \hat{i}+2\hat{j}-7\hat{k}\right)=0 \\ \Rightarrow \lambda +4-21=0 \\ \Rightarrow \lambda -17=0 \\ \Rightarrow \lambda =17 \end{gathered}$$

$$\begin{gathered} \left(ii\right)\text{We know that the planes}{a}_{1}x+b_{1}y+c_{1}z+d_1=0\text{and}{a}_{2}x+b_{2}y+c_{2}z+d_2=0\text{are}\text{perperndicular to each other only if} \\ {a}_1{a}_2+b_1{b}_2+c_1{c}_2=0 \\ \text{The given planes are 2}x-4y+3z=5\text{and}x+2y+\lambda z=5. \\ \Rightarrow a_1=2;b_1=-4;c_1=3;a_2=1;b_2=2;c_2=\lambda \mathit{} \\ \text{It is given that the given planes are perpendicular.} \\ \Rightarrow a_1{a}_2+b_1{b}_2+c_1{c}_2=0 \\ \Rightarrow \left(2\right)\left(1\right)+\left(-4\right)\left(2\right)+\left(3\right)\left(\lambda \right)=0 \\ \Rightarrow 2-8+3\lambda =0 \\ \Rightarrow 3\lambda =6 \\ \Rightarrow \lambda =2 \end{gathered}$$

$$\begin{gathered} \left(iii\right)\text{We know that the planes}{a}_{1}x+b_{1}y+c_{1}z+d_1=0\text{and}{a}_{2}x+b_{2}y+c_{2}z+d_2=0\text{are}\text{perperndicular to each other only if} \\ {a}_1{a}_2+b_1{b}_2+c_1{c}_2=0 \\ \text{The given planes are 3}x-6y-2z=7\text{and}2x+y-\lambda z=5. \\ \Rightarrow a_1=3;b_1=-6;c_1=-2;a_2=2;b_2=1;c_2=-\lambda \mathit{} \\ \text{The given planes are perpendicular.} \\ \Rightarrow a_1{a}_2+b_1{b}_2+c_1{c}_2=0 \\ \Rightarrow \left(3\right)\left(2\right)+\left(-6\right)\left(1\right)+\left(-2\right)\left(-\lambda \right)=0 \\ \Rightarrow 6-6+2\lambda =0 \\ \Rightarrow 2\lambda =0 \\ \Rightarrow \lambda =0 \end{gathered}$$