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  • Explain solution RD Sharma class 12 Chapter 28 The Plane exercise 28.11 question 4

Explain solution RD Sharma class 12 Chapter 28 The Plane exercise 28.11 question 4

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Answer:  m = -3

Hint: Use properties of vector

Given: Line \overrightarrow{\mathrm{r}}=\hat{\mathrm{i}}+\lambda(2 \hat{\mathrm{i}}-\mathrm{m} \hat{\mathrm{j}}-3 \hat{\mathrm{k}})  and

            Plane \vec{r} \cdot(m \hat{i}+3 \hat{j}+\hat{k})=4

Solution: The given line is parallel to the vector  \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}-\mathrm{m} \hat{\mathrm{j}}-3 \hat{\mathrm{k}}  and the given plane is normal to the vector \overrightarrow{\mathrm{n}}=m \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}

If the line is parallel to the plane, the normal to the plane is perpendicular to the line

\begin{aligned} &\Rightarrow \vec{b} \perp \vec{n} \\ &\Rightarrow \vec{b} \cdot \vec{n}=0 \\ &\Rightarrow(2 \hat{i}-m \hat{j}-3 \hat{k})(m \hat{i}+3 \hat{j}+\hat{k})=0 \\ &\Rightarrow 2 m-3 m-3=0 \\ &\Rightarrow-m-3=0 \\ &\Rightarrow m=-3 \end{aligned}

 

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