A line (\(1\)-dimensional object) in space may be described using a point on the line and a single direction vector.

It naturally follows that a plane (\(2\)-dimensional object) can be described using a point in the plane and two direction vectors.

However, recall that two collinear vectors lie on (and describe) the same line; thus, the two direction vectors describing the plane must be non-collinear.

The vector equation of a plane describes the position vector, \(\overrightarrow{OP}\), of any arbitrary point, \(P\), in the plane.

To derive the vector equation for an arbitrary plane, consider the following:

Let \( P_0 \) be a fixed point in the plane, and let \( P \) be any other arbitrary point in the plane.

The coordinates of \( P \) can be determined from the vector sum

as shown in the diagram.

If we take any two non-collinear vectors in the plane, then any arbitrary vector in the plane (such as \( \overrightarrow{P_0P} \)) can be expressed as a vector sum of scalar multiples of these two vectors.

Let \( \vec{a} \) and \( \vec{b} \) be two such vectors in the plane.

Then \( s\vec{a} + t\vec{b} = \overrightarrow{P_0P}\), for some \(s,t \in \mathbb{R} \) as shown in the diagram.

This vector \( s\vec{a} + t\vec{b}\) is called a linear combination of vectors \(\vec{a}\) and \(\vec{b}\).

Substituting this into equation \((1)\), \(\overrightarrow{OP} = \overrightarrow{OP_0} + \overrightarrow{P_0P}\), we get \(\overrightarrow{OP} = \overrightarrow{OP_0} + s\vec{a} + t\vec{b},~~s,t \in \mathbb{R}\) which is the vector equation of the plane.

Rewriting the vector equation of a plane into its \(x\), \(y\), and \(z\) components, we get