Find the equation of the plane passing through the intersection of the planes and
and parallel to x-axis. Hence .find the distance of the plane from x-axis.
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Equation of plane passing through the intersection of the planes
![\left [ \vec{r} \cdot \left ( \hat{i}+\hat{j} +\hat{k}\right )-1\right ]+\lambda \left [ \vec{r}\cdot \left ( 2\hat{i}+3\hat{j} -\hat{k} \right )+4 \right ]= 0](https://learn.careers360.com/latex-image/?%5Cleft%20%5B%20%5Cvec%7Br%7D%20%5Ccdot%20%5Cleft%20%28%20%5Chat%7Bi%7D+%5Chat%7Bj%7D%20+%5Chat%7Bk%7D%5Cright%20%29-1%5Cright%20%5D+%5Clambda%20%5Cleft%20%5B%20%5Cvec%7Br%7D%5Ccdot%20%5Cleft%20%28%202%5Chat%7Bi%7D+3%5Chat%7Bj%7D%20-%5Chat%7Bk%7D%20%5Cright%20%29+4%20%5Cright%20%5D%3D%200)
![\vec{r}\cdot \left [ \left ( 1+2\lambda \right )\hat{i}+\left ( 1+3\lambda \right )\hat{j}+\left ( 1-\lambda \right )\hat{k} \right ] = 1+4\lambda = 0\: ---(1)](https://learn.careers360.com/latex-image/?%5Cvec%7Br%7D%5Ccdot%20%5Cleft%20%5B%20%5Cleft%20%28%201+2%5Clambda%20%5Cright%20%29%5Chat%7Bi%7D+%5Cleft%20%28%201+3%5Clambda%20%5Cright%20%29%5Chat%7Bj%7D+%5Cleft%20%28%201-%5Clambda%20%5Cright%20%29%5Chat%7Bk%7D%20%5Cright%20%5D%20%3D%201+4%5Clambda%20%3D%200%5C%3A%20---%281%29)
![\vec{r}\cdot \left [ \left ( \hat{i}+3\left ( \frac{-1}{2} \right )\hat{j} +\hat{k}\left ( 1-\left ( \frac{-1}{2} \right ) \right )\right ) \right ]](https://learn.careers360.com/latex-image/?%5Cvec%7Br%7D%5Ccdot%20%5Cleft%20%5B%20%5Cleft%20%28%20%5Chat%7Bi%7D+3%5Cleft%20%28%20%5Cfrac%7B-1%7D%7B2%7D%20%5Cright%20%29%5Chat%7Bj%7D%20+%5Chat%7Bk%7D%5Cleft%20%28%201-%5Cleft%20%28%20%5Cfrac%7B-1%7D%7B2%7D%20%5Cright%20%29%20%5Cright%20%29%5Cright%20%29%20%5Cright%20%5D)
![\Rightarrow\vec{r}\cdot \left [ -\frac{1}{2}\hat{j}+\frac{3}{2}\hat{k} \right ]= 3](https://learn.careers360.com/latex-image/?%5CRightarrow%5Cvec%7Br%7D%5Ccdot%20%5Cleft%20%5B%20-%5Cfrac%7B1%7D%7B2%7D%5Chat%7Bj%7D+%5Cfrac%7B3%7D%7B2%7D%5Chat%7Bk%7D%20%5Cright%20%5D%3D%203)
![\Rightarrow\vec{r}\cdot \left [ -\hat{j}+3\hat{k} \right ]= 6](https://learn.careers360.com/latex-image/?%5CRightarrow%5Cvec%7Br%7D%5Ccdot%20%5Cleft%20%5B%20-%5Chat%7Bj%7D+3%5Chat%7Bk%7D%20%5Cright%20%5D%3D%206)
Direction ration (1,0,0)
Equation (1)
Distance of the point from x axis: x(x,0,0)