Find the Cartesian equation of the following planes:

(a) $\vec r \cdot (\hat i + \hat j - \hat k) = 2$
(b) $\vec r \cdot (2\hat i + 3\hat j - 4\hat k) = 1$
(c)$\vec r \cdot [(s - 2t)\hat i + (3 - t)\hat j + (2s + t)\hat k] = 15$,

.: (a) We have, $\vec r \cdot (\hat i + \hat j - \hat k) = 2$
$\Rightarrow$ $(x\hat i + y\hat j + z\hat k) \cdot (\hat i + \hat j - \hat k) = 2$
$\Rightarrow$ $x + y - z = 2,$

which is the required cartesian equation.

(b) We have, $\vec r \cdot (2\hat i + 3\hat j - 4\hat k) = 1$

$\Rightarrow$ $(x\hat i + y\hat j + z\hat k) \cdot (2\hat i + 3\hat j - 4\hat k) = 1 \Rightarrow 2x + 3y - 4z = 1$

which is the required cartesian equation.

(c) We have, $\vec r \cdot [(s - 2t)\hat i + (3 - t)\hat j + (2s + t)\hat k] = 15$

$\Rightarrow$ $(x\hat i + y\hat j + z\hat k) \cdot [(s - 2t)\hat i + (3 - t)\hat j + (2s + t)\hat k = 15$

$\Rightarrow$ $(s - 2t)x + (3 - t)y + (2s + t)z = 15,$

which is the required cartesian equation.