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  • Provide solution for RD Sharma maths class 12 chapter The Plane exercise 28.3 question 18

Provide solution for RD Sharma maths class 12 chapter The Plane exercise 28.3  question 18

Answers (1)

Answer:

The answer of given question is 2x+3y-z=20

Hint:

By using formula \vec{A} . \vec{B}=A_{x} B_{x}+A_{y} B_{y}+A_{z} B_{z}

Given:

The plane is passing through P(5,2,-4) and perpendicular to the line having 2,3,-1 as the direction ratio.

Solution:

Let, the position vector of this point P be

\Rightarrow \vec{a}=5 \hat{\imath}+2 \hat{\jmath}-4 \hat{k} \ldots(i)

And also given the plane is normal having 2,3,-1

As the direction ratio.

\vec{n}=2 \hat{\imath}+3 \hat{\jmath}-\hat{k} \ldots(ii)

We know that

(\vec{r}-\vec{a}) \cdot \vec{n}=0

Substituting the values from equation (i) and equation (ii) in the above equation we get

\begin{aligned} &{[\vec{r}-(5 \hat{\imath}+2 \hat{\jmath}-4 \hat{k})] \cdot(2 \hat{\imath}+3 \hat{\jmath}-\hat{k})=0} \\ &\Rightarrow \vec{r} \cdot(2 \hat{\imath}+3 \hat{\jmath}-\hat{k})-(5 \hat{\imath}+2 \hat{\jmath}-4 \hat{k}) \cdot(2 \hat{\imath}+3 \hat{\jmath}-\hat{k})=0 \\ &\Rightarrow \vec{r} \cdot(2 \hat{\imath}+3 \hat{\jmath}-\hat{k})-[(2)(5)+(2)(3)+(-4)(-1)]=0 \end{aligned}

By multiplying the two vectors using the formula

\begin{aligned} &\vec{A} \cdot \vec{B}=A_{x} B_{x}+A_{y} B_{y}+A_{z} B_{z} \\ &\Rightarrow \vec{r} \cdot(2 \hat{\imath}+3 \hat{\jmath}-\hat{k})-[10+6+4]=0 \\ &\Rightarrow \vec{r} \cdot(2 \hat{\imath}+3 \hat{\jmath}-\hat{k})-20=0 \end{aligned}

\Rightarrow \vec{r} \cdot(2 \hat{\imath}+3 \hat{\jmath}-\hat{k})=20 is the vector equation of a required plane.

\vec{r}=(x \hat{\imath}+y \hat{\jmath}+z \hat{k})

Then the above vector equation of a plane becomes,

(x \hat{\imath}+y \hat{\jmath}+z \hat{k}) \cdot(2 \hat{\imath}+3 \hat{\jmath}-\hat{k})=20

Now multiplying the two vectors using the formula

\begin{aligned} &\vec{A} . \vec{B}=A_{x} B_{x}+A_{y} B_{y}+A_{z} B_{z} \\ &\Rightarrow(x)(2)+(y)(3)+(z)(-1)=20 \\ &\Rightarrow 2 x+3 y-z=20 \end{aligned}

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

 

Posted by

infoexpert26

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