Q.Find the vector equation of the plane passing through the point having position vector $2\hat i+3\hat j+4\hat k$ and perpendicular to the vector $2\hat i+\hat j-2\hat k$.

A plane is completely fixed once you know one of its points, $A(\vec a)$, and one vector $\vec n$ perpendicular to it (its normal). For any other point $P(\vec r)$ of the plane, the chord $\overrightarrow{AP} = \vec r - \vec a$ lies flat in the plane, so it must be perpendicular to the normal: $(\vec r - \vec a)\cdot\vec n = 0$. Expanding the dot product gives the vector equation in scalar-product form, $\vec r\cdot\vec n = \vec a\cdot\vec n$ — often abbreviated to $\vec r\cdot\vec n = d$ once the constant $\vec a\cdot\vec n$ is called $d$. …