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Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane \vec{r} \cdot(\hat{i}+2 \hat{j}-5 \hat{k})+9=0
CBSE | Class 12 | Maths | Miscellaneous Exercise | NCERT | Three Dimensional Geometry
Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane \vec{r} \cdot(\hat{i}+2 \hat{j}-5 \hat{k})+9=0

Solution:

The vector eq. of a line passing through a point with position vector \vec{a} and parallel to vector \vec{b} is given as
\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}}
Given that the line passes through (1,2,3)
Therefore, \vec{a}=1 \hat{i}+2 \hat{j}+3 \hat{k}
Let’s find the normal of plane
\begin{array}{l} \overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})+9=0 \\ \overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})=-9 \\ -\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})=9 \\ \overrightarrow{\mathrm{r}} \cdot(-1 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})+9=0 \end{array}
Now compare it with \overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{n}}=\mathrm{d}
\overrightarrow{\mathrm{n}}=-\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}
As line is perpendicular to plane, the line will be parallel of the plane
\therefore \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{n}}=-\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}
Therefore,
\begin{array}{l} \overrightarrow{\mathrm{r}}=(1 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\lambda(-\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}) \\ \overrightarrow{\boldsymbol{r}}=(1 \hat{\boldsymbol{i}}+2 \hat{\boldsymbol{j}}+3 \hat{\boldsymbol{k}})-\lambda(\hat{\boldsymbol{i}}+2 \hat{\boldsymbol{j}}-5 \hat{\boldsymbol{k}}) \end{array}
As a resullt, the required vector equation of line is \vec{r}=(1 \hat{\boldsymbol{i}}+2 \hat{\boldsymbol{j}}+3 \hat{\boldsymbol{k}})-\lambda(\hat{\boldsymbol{i}}+2 \hat{\boldsymbol{j}}-5 \hat{\boldsymbol{k}})

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