Three ways of describing a line in \(\mathbb{R}^2\) have been discussed so far:

Vector Equation

Parametric Equations

Scalar Equation

There is a natural extension of the vector and parametric equation descriptions to lines in \(\mathbb{R}^3\).

However, the scalar equation does not generalize as it is defined by the normal vector to a line.

Since there are infinitely many normals to a given line in \(3\) dimensions, there is no valid definition.

Similar to the case in \(2\) dimensions, the vector equation of a line in \(\mathbb{R}^3\) can be described using a point on the line and a direction vector for the line.

Let \( P_0~(x_0, y_0, z_0) \) be a specific point on the line.

Let \( P~(x, y, z) \) be an arbitrary point on the line.

Using the triangle law of vector addition,

where \(\vec{d}=(a,b,c)\) is a direction vector for the line.

This is the vector equation of a line in \(\mathbb{R}^3\).

Considering the individual components of the vector equation of a line in \(3\)-space gives the parametric equations

where \( t \in \mathbb{R} \) and \( \vec{d} = (a, b, c) \) is a direction vector of the line.

Using the three parametric equations and rearranging each to solve for \(t\), gives the symmetric equations of a line in \(\mathbb{R}^3\).