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Please solve RD Sharma class 12 chapter The Plane exercise 28.3  question 21 maths textbook solution

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Answer:

The answer of given question is \vec{r} \cdot(4 \hat{\imath}-3 \hat{\jmath}+6 \hat{k})=12

Hint:

Let the equation be Ax+By+Cz+D=0

Given:

Plane with intercepts 3,-4 and 2 on x, y and z axes

Solution:

Let the equation be Ax+By+Cz+D=0….(i)

Let the plane meets the  x,y,z axes (3,0,0), (0,-4,0) and (0,0,2) respectively.

Putting (0,0,2), we get

\begin{aligned} &A(0)+B(0)+C(2)+D=0 \\ &\Rightarrow C=-\frac{D}{2} \end{aligned}

Putting (0,-4,0), we get

\begin{aligned} &A(0)+B(-4)+C(0)+D=0 \\ &\Rightarrow B=\frac{D}{4} \end{aligned}

And putting (3,0,0) we get

\begin{aligned} &A(3)+B(0)+C(0)+D=0 \\ &\Rightarrow A=-\frac{D}{3} \end{aligned}

Substituting the values of A, B, C in equation (i) we get by putting B(0,-4,0), we get

\begin{aligned} &\left(-\frac{D}{3}\right) x+\frac{D}{4} y+\left(-\frac{D}{2}\right) z+D=0 \\ &\Rightarrow \frac{-4 D x+3 D y-6 D z+12 D}{2}=0 \\ &\Rightarrow(-D)(4 x-3 y+6 z-12)=0 \\ &\Rightarrow 4 x-3 y+6 z=12 \end{aligned}

This is the cartesian form of equation of the required plane.

Now vector equation form of the plane4x-3y+6z=12

\begin{aligned} &(x \hat{\imath}+y \hat{\jmath}+z \hat{k}) \cdot(4 \hat{\imath}-3 \hat{\jmath}+6 \hat{k})=12 \\ &\Rightarrow \vec{r} \cdot(4 \hat{\imath}-3 \hat{\jmath}+6 \hat{k})=12 \end{aligned}

 

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