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Provide solution for RD Sharma maths class 12 chapter The Plane exercise 28.3  question 13 sub question (ii)

Answers (1)

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Answer:

The normal to the given pairs of planes are perpendicular to each other.

Hint:

\overrightarrow{n_{1}} \cdot \overrightarrow{n_{2}}=0

Given:

\vec{r} \cdot(2 \hat{\imath}-\hat{\jmath}+3 \hat{k})=5 \text { and } \vec{r} \cdot(2 \hat{\imath}-2 \hat{\jmath}-2 \hat{k})=5

Solution:

The equation of the first plane is

\vec{r} \cdot(2 \hat{\imath}-\hat{\jmath}+3 \hat{k})=5

The normal to this plane is

\overrightarrow{n_{1}}=2 \hat{\imath}-\hat{\jmath}+3 \hat{k} \ldots(i)

The equation of the second plane is

\vec{r} \cdot(2 \hat{\imath}-2 \hat{\jmath}-2 \hat{k})=5

The normal to this plane is

\overrightarrow{n_{2}}=2 \hat{\imath}-2 \hat{\jmath}-2 \hat{k} \ldots(ii)
Now,

\begin{aligned} &\overrightarrow{n_{1}} \cdot \overrightarrow{n_{2}}=(2 \hat{\imath}-\hat{\jmath}+3 \hat{k}) \cdot(2 \hat{\imath}-2 \hat{\jmath}-2 \hat{k}) \\ &\Rightarrow \overrightarrow{n_{1}} \cdot \overrightarrow{n_{2}}=(2)(2)+(-1)(-2)+(3)(-2)=0 \end{aligned}

Hence \overrightarrow{n_{1}} is perpendicular to \overrightarrow{n_{2}}

Therefore, the normal to the given pair of planes are perpendicular to each other.

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