Straight Lines in ThreeDimensional Geometry

What Is a Straight Line in Space?

In threedimensional Euclidean space a straight line is the set of points that satisfy a linear relationship among the three coordinates. Geometrically it is the most elementary onedimensional object that can be drawn in space. Every line can be described by a point through which it passes and a direction that tells how one moves from that point.

Unlike a line in the plane, a line in space may be parallel to a plane, intersect a plane in a single point, be contained in a plane, or be skew (nonparallel and nonintersecting) with another line. Understanding the algebraic descriptions of a line is essential for solving problems that involve distances, angles, and intersections.

Vector Equation of a Line

Let \(\mathbf{r}\) be the position vector of an arbitrary point \(P(x,y,z)\) on the line, \(\mathbf{a}\) the position vector of a fixed point \(A(x_0,y_0,z_0)\) on the line, and \(\mathbf{d}=\langle l,m,n\rangle\) a direction vector that gives the lines orientation. The vector equation is

\[\mathbf{r}= \mathbf{a}+t\,\mathbf{d},\qquad t\in\mathbb{R}\]    

The parameter \(t\) measures how far we travel along the direction vector from the initial point. When \(t=0\) we are at \(A\); when \(t=1\) we reach the point \(A+\mathbf{d}\); negative values of \(t\) move in the opposite direction.

Parametric Form

By separating the components of the vector equation we obtain three linear equations in the single parameter \(t\):

\[\begin{cases}x = x_{0}+lt\\[4pt]y = y_{0}+mt\\[4pt]z = z_{0}+nt\end{cases}\qquad (t\in\mathbb{R})\]    

This form is particularly convenient for substituting the line into plane equations, because each coordinate is expressed directly in terms of the same parameter.

Symmetric Form

If the direction components \(l,m,n\) are all nonzero we may eliminate the parameter \(t\) to write the line in symmetric form:

\[\frac{x-x_0}{\,l\,}= \frac{y-y_0}{\,m\,}= \frac{z-z_0}{\,n\,}\]    

The three fractions are equal to the same number \(t\). When one or more of the direction components are zero, the corresponding coordinate remains constant. For instance, if \(n=0\) the line is parallel to the \(xy\)-plane and the equation becomes \[ \frac{x-x_0}{l}= \frac{y-y_0}{m},\qquad z=z_0. \]

Intersection with a Plane

A plane in \(\mathbb{R}^3\) can be written in the general form \[ Ax+By+Cz+D=0. \] To find where a line meets the plane (if it does), substitute the parametric expressions for \(x,y,z\) into the plane equation and solve for the parameter \(t\). The resulting value of \(t\) gives the intersection point.

If the substitution results in an identity (e.g., \(0=0\)) the line lies entirely in the plane; if it yields a contradiction (e.g., \(0=7\)) the line is parallel to the plane and does not intersect it.

Distance Between Skew Lines

Two lines are skew when they are not parallel and do not intersect. The shortest distance between them is the length of the common perpendicular. Suppose the lines are \[ L_1:\;\mathbf{r}= \mathbf{a}+s\mathbf{d},\qquad L_2:\;\mathbf{r}= \mathbf{b}+t\mathbf{e}, \] where \(\mathbf{d}\) and \(\mathbf{e}\) are direction vectors. The vector \(\mathbf{b}-\mathbf{a}\) joins any point on \(L_1\) to any point on \(L_2\). The distance \(D\) is given by

\[D=\frac{\bigl|(\mathbf{b}-\mathbf{a})\cdot(\mathbf{d}\times\mathbf{e})\bigr|}           {\|\mathbf{d}\times\mathbf{e}\|}.\]    

The numerator is the absolute value of the scalar triple product; the denominator is the magnitude of the cross product of the two direction vectors, which equals the area of the parallelogram spanned by \(\mathbf{d}\) and \(\mathbf{e}\).

Angle Between Two Lines

The acute angle \(\theta\) between two nonparallel lines with direction vectors \(\mathbf{d}_1=\langle l_1,m_1,n_1\rangle\) and \(\mathbf{d}_2=\langle l_2,m_2,n_2\rangle\) is defined by the dot product formula

\[\cos\theta=\frac{\mathbf{d}_1\cdot\mathbf{d}_2}                 {\|\mathbf{d}_1\|\,\|\mathbf{d}_2\|},\qquad 0\le\theta\le\frac{\pi}{2}.\]    

The same formula works when the lines intersect, and it also gives the angle between any two parallel lines (which is \(0\) degrees). If the lines are skew, the angle obtained this way is the angle between their direction vectors, not the angle between the skew lines themselves (the latter is not defined in the usual sense).

Applications and Further Topics

Straight lines in three dimensions appear in many scientific and engineering contexts:

• Ray tracing in computer graphics each ray is a line emanating from a camera.
• Structural analysis beams and columns are modelled as line segments.
• Navigation the shortest path between two points in space is a straight line.
• Robotics the motion of a joint often follows a line, and inversekinematic equations use line representations.
• Physics the trajectory of a particle moving at constant velocity is linear.

More advanced subjects build on the concepts presented here:

• Lines and planes in vector spaces extending to higher dimensions.
• Projective geometry treating points at infinity as intersecting directions.
• Analytic geometry of conics and quadrics where lines often serve as asymptotes.
• Computeraided design (CAD) where geometric constraints are expressed using line equations.