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  • Explain solution RD sharma class 12 chapter 28 The Plane exercise 28.8, question 17

Explain solution RD sharma class 12 chapter 28 The Plane exercise 28.8, question 17

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The answer of the given question is x+y+z=a+b+c.

Hints:-  By substituting \vec{r}=x\hat{i}+y\hat{j}+z\hat{k}  in equation (ii)

Given:-  (a, b, c) and parallel to plane  \vec{r}.\left (\hat{i}+\hat{j}+\hat{k} \right )=2                      

Solution:-  Any plane passes through the point (a, b, c) and parallel to plane  \vec{r}.\left (\hat{i}+\hat{j}+\hat{k} \right )=2 is given by

\vec{r}.\left (\hat{i}+\hat{j}+\hat{k} \right )=\lambda…(i)

Here, the position vector  \vec{r} of this point is \vec{r}=a\hat{i}+b\hat{j}+c\hat{k}

∴ Equation (i) becomes
\left (a\hat{i}+b\hat{j}+c\hat{k} \right ).\left ( \hat{i}+\hat{j}+\hat{k} \right )=\lambda \\ \Rightarrow a+b+c=\lambda

Substituting \lambda = a+b+c in equation (i), we obtain

r.i +j+k=a+b+c                                … (ii)

This is vector equation of the required plane

Substituting \vec{r}=x\hat{i}+y\hat{j}+z\hat{k} in equation (ii)

\left (x\hat{i}+y\hat{j}+z\hat{k} \right ).\left ( \hat{i}+\hat{j}+\hat{k} \right )=a+b+c\\ x+y+z=a+b+c

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