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  • Need solution for RD Sharma maths class 12 chapter 28 The Plane exercise 28.6 question 3 sub question 1 maths textbook solution

Need solution for RD Sharma maths class 12 chapter 28 The Plane exercise 28 point 6 question 3 sub question 1 maths textbook solution

Answers (1)

Answer:

              We need to show that the following planes are at right angle

Hint:

              \cos \theta=\frac{\vec{n}_{1} \cdot \vec{n}_{2}}{\left|\vec{n}_{1}\right| \cdot\left|\vec{n}_{2}\right|}=0

Given:

              \overrightarrow{\boldsymbol{r}} \cdot(2 \hat{\boldsymbol{\imath}}-\hat{\boldsymbol{j}}+\widehat{\boldsymbol{k}})=\mathbf{5} \text { And } \vec{r} \cdot(-\hat{\imath}-\hat{\jmath}+\hat{k})=3

Solution:

Here

\begin{aligned} &\vec{n}_{1}=2 \hat{\imath}-\hat{\jmath}+\hat{k}, \text { on comparing with } \overrightarrow{\boldsymbol{r}} \cdot(\mathbf{\imath} \hat{\boldsymbol{\imath}}-\hat{\boldsymbol{j}}+\widehat{\boldsymbol{k}})=\mathbf{5}\\ &\vec{n}_{2}=-\hat{\imath}-\hat{\jmath}+\widehat{k} \text { on comparing wit } \vec{r} \vec{r} \cdot(-\hat{\imath}-\hat{\jmath}+\hat{k})=3\\ &\cos \theta=\frac{\vec{n}_{1} \cdot \vec{n}_{2}}{\left|\vec{n}_{1}\right| \cdot\left|\vec{n}_{2}\right|}\\ &=\frac{(2 \hat{\imath}-\hat{\jmath}+\hat{k}) \cdot(-\hat{\imath}-\hat{\jmath}+\hat{k})}{|2 \hat{\imath}-\hat{\jmath}+\hat{k}| \cdot|-\hat{\imath}-\hat{\jmath}+\hat{k}|}\\ &=\frac{-2+1+1}{\sqrt{(2)^{2}+(-1)^{2}+(1)^{2}} \cdot \sqrt{(-1)^{2}+(-1)^{2}+(1)^{2}}}\\ &=0 \end{aligned}

\theta =\frac{\pi }{2}  .......hence proved

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