Equations of lines in 3D (HL)

Direction vectors are analogous to gradients.

In vector geometry, a line is identified by a point $(x_0, y_0, z_0)$ on the line, and a direction vector $(l, m, n)$ of the line.

The direction vector can be replaced by any non-zero scalar multiple. The point can be replaced by any other point on the line.

Contents

• Vector equation of a line
• Parametric equation of a line
• Symmetric (Cartesian) equation of a line
• Checking if a point is on a line
• Finding equation of a line.
• Applications
• Tips

Vector equation of a line

The position vector $\bm r$ of any point on the line is

The vector multiplied by a parameter $\lambda$ is the direction vector.

Different points on the line are associated with different parameters.

Parametric equation of a line

Symmetric (Cartesian) equation of a line

This solves for $\lambda$ and equate the expressions.

Where the component of the direction vector is $0$, for example if direction vector is $(2, 0, -3)$, then

Note, when there is not a denominator, it is $1$. Beware of potential of a reversed numerator, because of multiplying by $-1$.

Checking if a point is on a line

Suppose we have

We want to verify if $(-7, -2, 6)$ is on the line.

If the direction vector has all non-zero components, then the parameter must be the same value for $x$, $y$, $z$.

So $(-7, -2, 6)$ is not on the line. Note that we simply substituted values into the symmetric (Cartesian) equation of a line.

Finding equation of a line.

If given two points $(x_0, y_0, z_0)$ and $(x_1, y_1, z_1)$, the equation of the line is

Namely, find a direction vector as the difference between two points, then include one of the given points (position vectors).

Applications

The equations of a 3D line can be interpreted to represent the path of some object or particle, with initial position $(x_0, y_0, z_0)$, constant velocity $(l, m, n)$, time traveled $\lambda$, and final position $\bm r$.

Tips

1. All forms of a line (using vectors) are accepted. Vector equation is probably the easiest.
2. Use different parameters for different lines.