Question

Find the equation of the plane passing through (a, b, c) and parallel to the plane $r.(\hat{i}+\hat{j}+\hat{k})=2$

Answer 1

Given plane is $r.(\hat{i}+\hat{j}+\hat{k})=2$ . . . (i)
The equation of the plane parallel to plane (i) is
$r.(\hat{i}+\hat{j}+\hat{k})=d$
or $(x\hat{i}+y\hat{j}+z\hat{k}).(\hat{i}+\hat{j}+\hat{k})=d$ . . . (ii)
Since, it passes through point (a, b, c).
$\therefore (a\hat{i}+b\hat{j}+z\hat{k}).(\hat{i}+\hat{j}+\hat{k})=d\Rightarrow a+b+c=d$
Putting $r=x\hat{i}+y\hat{j}+z\hat{k}$ and d = a + b + c in Eq. (ii), we have
$(x\hat{i}+y\hat{j}+z\hat{k}).(\hat{i}+\hat{j}+\hat{k})=a+b+c\Rightarrow x+y+z=a+b+c$
Which is required equation of the plane.