Find the equation of the plane which contains the line of intersection of the planes vector (r) . (i -2j + 3k) - 4 = 0)

Question

Find the equation of the plane which contains the line of intersection of the planes \(\vec{r} .(\hat{i}-2\hat{j}+3\hat{k})-4=0\) and \(\vec{r} .(-2\hat{i}+\hat{j}+\hat{k})+5=0\) and whose intercept on x-axis is equal to that of on y-axis.

Answer 1

Given planes are \(\vec{r} .(\hat{i}-2\hat{j}+3\hat{k})-4=0\) and \(\vec{r} .(-2\hat{i}+\hat{j}+\hat{k})+5=0\)

These can be written in cartesian form as

Now the equation of plane containing the line of intersection of the planes (i) and (ii) is given by

Putting the value of \(\lambda = 1\) in (iii), we get

\(\therefore\) Its vector form is