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The position vector of a point $P$ is $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{a}=-\hat{\mathbf{i}}-2 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are two vectors which determine a plane $\pi$. The equation of a line through $P$ normal to $\mathbf{b}$ and lying on the plane $\pi$ is
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Verified Answer
The correct answer is:
$r=2 \hat{i}+\hat{j}+3 \hat{k}+\lambda(-\hat{i}+5 \hat{j}-2 \hat{k})$
Given,
Position vector of a point $P$ is $(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}})$
Equation of plane,
$(\mathbf{r}-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \cdot(\mathbf{a} \times \mathbf{b})=0$
$\therefore$ Normal of plane $=\mathbf{a} \times \mathbf{b}$
Equation of line through $\mathbf{P}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ and normal to $\mathbf{b}$ and lying on the plane is
$\begin{aligned}
& \mathbf{r}=(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda((\mathbf{a} \times \mathbf{b}) \times \mathbf{b}) \\
& \mathbf{r}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda((\mathbf{b} . \mathbf{a}) \mathbf{b}-(\mathbf{b} . \mathbf{b}) \mathbf{a}) \\
& \mathbf{r}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda[(-5)(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})-6(-\hat{\mathbf{i}}-2 \hat{\mathbf{k}})] \\
& \mathbf{r}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda(-\hat{\mathbf{i}}+5 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})
\end{aligned}$
Position vector of a point $P$ is $(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}})$
Equation of plane,
$(\mathbf{r}-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \cdot(\mathbf{a} \times \mathbf{b})=0$
$\therefore$ Normal of plane $=\mathbf{a} \times \mathbf{b}$
Equation of line through $\mathbf{P}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ and normal to $\mathbf{b}$ and lying on the plane is
$\begin{aligned}
& \mathbf{r}=(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda((\mathbf{a} \times \mathbf{b}) \times \mathbf{b}) \\
& \mathbf{r}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda((\mathbf{b} . \mathbf{a}) \mathbf{b}-(\mathbf{b} . \mathbf{b}) \mathbf{a}) \\
& \mathbf{r}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda[(-5)(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})-6(-\hat{\mathbf{i}}-2 \hat{\mathbf{k}})] \\
& \mathbf{r}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda(-\hat{\mathbf{i}}+5 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})
\end{aligned}$
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