Question

Find the Cartesian equation of the following plane:

$r.\left[\right(s-2t)\hat{i}+(3-t)\hat{j}+(2s+t\left)\hat{k}\right]=15$

Answer 1

Given equation is $r.\left[\right(s-2t)\hat{i}+(3-t)\hat{j}+(2s+t\left)\hat{k}\right]=15$ ...(iii)

For any arbitrary point P(x,y,z) on the plane, position vector r is given by, $x\hat{i}+y\hat{j}+z\hat{k}$

Substituting the value of r in Eq. (iii), we obtain

$(x\hat{i}+y\hat{j}+z\hat{k}).\left\{\right(s-2t)\hat{i}+(3-t)\hat{j}+(2s+t\left)\hat{k}\right\}=15\Rightarrow x(s-2t)+y(3-t)+z(2s+t)-15=0$

This is the Cartesian equation of the given plane.