Equation of Line in 3D

Position and Direction Vectors

A vector is a mathematical object that has both direction and magnitude. They can be written in two forms:

• Column vector form: \( \begin{bmatrix} x \\ y \\ z \end{bmatrix}. \)

• Unit vector form: \( x \vec{i} + y \vec{j} + z \vec{k}. \)

Fig. 1. A vector can be thought of as an arrow, pointing from one place to another.

A position vector is a vector representing a point in space, just like coordinates, while a direction vector represents a movement. If you move by a direction vector from the origin, you will reach the point of its corresponding position vector.

Vectors can be added and subtracted by adding or subtracting the individual components. In column form, that means adding the first entries in each vector for the first entry in the new vector, and adding the second entries in each vector for the second entry in the new vector, and so on. In unit vector form, this just means adding and subtracting the like terms as you would with any other algebraic equation.

Vectors can also be multiplied by scalars, by multiplying each of the individual components by the scalar. For a column vector, this just means multiplying each entry by the scalar. For a vector in unit vector form, this just means expanding the brackets in the usual way. For more information, see Core Vectors.

Parametric Equation of a Line in 3D

The parametric equations for a straight line in 3 dimensions are:\[ \begin{align} x & = a_1 + t b_1 \\ y & = a_2 + t b_2 \\ z & = a_3 + t b_3 \end{align} \]

where

\[ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \]

is the position vector for a point on the line, and

\[\begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \]

is the direction vector of the line, and \(t\) is a scalar variable.

From this parametric definition, a vector form for a straight line in 3D can be obtained.

Vector Equation of a Line in 3D

If you define

\[ \vec{r} = \begin{bmatrix} x \\ y\\ z\end{bmatrix}, \vec{a} = \begin{bmatrix} a_1 \\ a_2 \\a_3 \end{bmatrix}, \vec{b} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \]

then the parametric form can be written in one simple vector equation, known as the vector form of a line in 3D:

\[ \vec{r} = \vec{a} + t \vec{b}, \]

which can be written in column vector form as:

\[ \begin{bmatrix} x \\ y\\ z \end{bmatrix} = \begin{bmatrix} a_1\\a_2\\a_3 \end{bmatrix} + t \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}. \]

Cartesian Equation of a Line in 3D

There is also a Cartesian Equation for a line in 3 dimensions. With the same parameters as for the parametric definition, the Cartesian Equation for a line in 3D is:

\[ \frac{ x - a_1}{b_1} = \frac{y-a_2}{b_2} = \frac{z-a_3}{b_3}. \]

This formula only works if the components of the direction vector are non-zero.

Equation of a Straight Line in 3D Passing Through Two Points

Given two points \( \vec{a} \) and \( \vec{b}, \) the vector form of a straight line between them will be:

\[ \vec{r} = \vec{a} + t (\vec{a} - \vec{b}). \]

This is because the direction vector between two position vectors is the subtraction of one of the vectors from the other. It doesn't matter which vector you use as the position vector, nor which order you take the subtraction in for the direction vector side. All this will affect is which values of \(t\) correspond to which points on the line.

Skew, Parallel and Intersecting Lines

In 2D space, if two different lines do not intersect, they must be parallel. In 3D space, it is possible for non-parallel lines to not intersect as well: these lines are called Skew lines.

Fig. 2. These 2 lines do not intersect but are also not parallel. This means they must be skew.

Above is an example of skewed lines. Two lines in 3D will be parallel if their direction vectors are scalar multiples of one another.

Fig. 3. These lines do not intersect but are going in the same direction, so they must be parallel.

To see if lines intersect, you can set them as equal to each other, giving three simultaneous equations in two variables. If all of these equations are solved for the same values, the lines intersect. If there are any contradictions, the lines do not intersect and must be parallel or skew.

Fig. 4. These lines intersect.

Let's see some examples, where you must determine whether lines are skewed, parallel, or intersecting. The first example gives the lines in Euclidean form.

This next example will give the lines in vector form.

Let's look at one final example of a question like this, this time with the line in parametric form.

Equation of a Line Parallel to the \(x\)-axis in 3D

To create a line parallel to the \(x\)-axis, you need a direction vector parallel to the \(x\)-axis. This is simply the standard unit vector, \(\vec{i}.\) Hence, for any position vector \( \vec{a},\) the vector equation for a line passing through \(\vec{a}\) that is parallel to the \(x\)-axis is:

\[ \vec{r} = \vec{a} + t \vec{i}.\]

The parametric equations of this line will be:

\[ \begin{align} x & = a_1 + t \\ y & = a_2 \\ z & = a_3. \end{align} \]

Since both the \(y \) and \(z\) components of the direction vector are \(0\), there is no Cartesian form of this line.

Equation of a Line Parallel to the \(z\)-axis in 3D

Finding a line parallel to the \(z\)-axis in 3D works exactly the same way as the line parallel to the \(x\)-axis from the previous section, but replacing \(\vec{i} \) with \(\vec{k}. \) Hence, for any position vector \( \vec{a},\) the vector equation for a line passing through \(\vec{a}\) that is parallel to the \(z\)-axis is:

\[ \vec{r} = \vec{a} + t \vec{k}.\]

The parametric equations of this line will be:

\[ \begin{align} x& = a_1 \\ y & = a_2 \\ z& = a_3+t. \end{align} \]

Since both the \(x \) and \(y\) components of the direction vector are \(0\), there is no Cartesian form of this line.