Equation of a Line in 3D

In two-dimensional geometry, a line is fully described by a single equation like y = mx + c. Moving into three-dimensional space, however, we need a richer framework because a line must be pinned down in all three coordinate directions simultaneously. The two most powerful tools for this are the vector form and the Cartesian (symmetric) form of a line's equation. The key idea is that any line in space is completely determined by two pieces of information: a point that lies on the line, and a direction that the line travels. The direction is captured by a direction vector, often written as \mathbf{b} = (b_1, b_2, b_3), while the known point is given by its position vector \mathbf{a} = (a_1, a_2, a_3). Every other point on the line can be reached by starting at \mathbf{a} and walking some scalar multiple of \mathbf{b}. Mastering lines in 3D is essential for studying planes, intersections, angles between lines, and countless applications in physics, engineering, computer graphics, and robotics. This article develops both the vector and Cartesian forms from first principles, shows how to convert between them, and works through concrete numerical examples.

Key takeaways
  • Start with a point and a direction vector
  • Write the vector form
  • Expand into component (parametric) form
  • Eliminate the parameter to get Cartesian (symmetric) form
  • Converting between forms

Start with a point and a direction vector

Let P_0 = (a_1, a_2, a_3) be a known point on the line, with position vector \mathbf{a}. Let \mathbf{b} = (b_1, b_2, b_3) be a non-zero direction vector parallel to the line. Any point P = (x, y, z) lies on the line if and only if the vector \overrightarrow{P_0P} = \mathbf{r} - \mathbf{a} is a scalar multiple of \mathbf{b}.

Write the vector form

The condition \mathbf{r} - \mathbf{a} = \lambda \mathbf{b} gives the vector equation \mathbf{r} = \mathbf{a} + \lambda \mathbf{b}, where \lambda \in \mathbb{R} is a real parameter. As \lambda varies over all real numbers, the position vector \mathbf{r} traces out every point on the line. This single compact equation replaces the three separate coordinate equations.

Expand into component (parametric) form

Writing \mathbf{r} = (x, y, z), \mathbf{a} = (a_1, a_2, a_3), and \mathbf{b} = (b_1, b_2, b_3), the vector equation becomes three scalar equations: x = a_1 + \lambda b_1,\quad y = a_2 + \lambda b_2,\quad z = a_3 + \lambda b_3. These are the parametric equations of the line. They are especially useful for finding specific points by substituting values of \lambda.

Eliminate the parameter to get Cartesian (symmetric) form

If all three direction components b_1, b_2, b_3 are non-zero, we can solve each parametric equation for \lambda and set them equal: \frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}. This is the Cartesian or symmetric form. It presents the line as the simultaneous solution of two independent equations. If one component, say b_3 = 0, the line is parallel to the xy-plane and the Cartesian form is written as \frac{x-a_1}{b_1} = \frac{y-a_2}{b_2},\ z = a_3.

Converting between forms

Vector → Cartesian: Read off \mathbf{a} and \mathbf{b} from the vector equation and substitute into the symmetric form. Cartesian → Vector: The denominators give the direction vector \mathbf{b} = (b_1, b_2, b_3) and the numerator shifts give the point \mathbf{a} = (a_1, a_2, a_3). Then write \mathbf{r} = \mathbf{a} + \lambda \mathbf{b}. Always check: any point satisfying the original conditions should verify the equation.

The Key Equation

r=a+λ b⟺x−a1b1=y−a2b2=z−a3b3\mathbf{r} = \mathbf{a} + \lambda\,\mathbf{b} \qquad \Longleftrightarrow \qquad \frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}
  • \mathbf{r} — Position vector of a general point (dimensionless (coordinates)): The vector (x, y, z) pointing from the origin to an arbitrary point P on the line.
  • \mathbf{a} — Position vector of a fixed point (dimensionless (coordinates)): The vector (a_1, a_2, a_3) pointing to a known point P_0 that lies on the line.
  • \lambda — Parameter (dimensionless scalar): A real number that varies from -\infty to +\infty. Different values of \lambda correspond to different points on the line.
  • \mathbf{b} — Direction vector (dimensionless (coordinates)): A non-zero vector (b_1, b_2, b_3) parallel to the line; it gives the direction in which the line travels through space.

Worked Example

Problem. A line passes through the point A = (2, -1, 3) and is parallel to the vector \mathbf{b} = (4, 1, -2). (a) Write the vector equation of the line. (b) Write the Cartesian (symmetric) equation. (c) Find the point on the line where \lambda = 3. (d) Determine whether the point Q = (10, 1, -1) lies on the line.

r=(2i^−j^+3k^)+λ(4i^+j^−2k^);x−24=y+11=z−3−2\mathbf{r} = (2\hat{i}-\hat{j}+3\hat{k})+\lambda(4\hat{i}+\hat{j}-2\hat{k}); \quad \frac{x-2}{4}=\frac{y+1}{1}=\frac{z-3}{-2}

Answer. Vector form: r = (2i − j + 3k) + λ(4i + j − 2k). Cartesian form: (x−2)/4 = (y+1)/1 = (z−3)/(−2). Point at λ=3 is (14, 2, −3). Q=(10,1,−1) lies on the line (λ=2).

Figure
Diagram
Line in 3D: Vector Form r = a + λbxzyOLine LA (fixed point)ab(direction vector)P (general point)rλ·br = a + λb (λ varies over all reals)Cartesian: (x−a₁)/b₁ = (y−a₂)/b₂ = (z−a₃)/b₃
Origins · 1 minRené Descartes and how this was discovered
RD
René Descartes
1596–1650 · French

Descartes invented the Cartesian coordinate system, published in his 1637 work La Géométrie. By assigning numerical coordinates to points in space, he united algebra and geometry, making it possible to write equations for geometric objects like lines and planes. The Cartesian (symmetric) form of a line in 3D bears his name directly. Descartes reportedly had the insight for his coordinate system while lying in bed watching a fly crawl across the ceiling — he realised he could describe the fly's position using two distances from two walls. Whether this story is literally true or not, it captures the essence of his great idea.

  • Equation of a Plane in 3D — A plane in 3D is another fundamental object described using a normal vector and a point, closely analogous to how a line uses a direction vector and a point.
  • Angle Between Two Lines in 3D — Once lines are expressed in vector form, the angle between them is found using the dot product of their direction vectors, a direct application of this topic.
  • Dot Product of Vectors — The dot product is essential for finding angles between direction vectors and for determining when two lines in 3D are perpendicular.
  • Skew Lines in Space — Skew lines are lines in 3D that neither intersect nor are parallel — a concept unique to three dimensions, analysed using the vector equations developed here.
  • Introduction to Vectors — The vector form of a line relies entirely on vector addition and scalar multiplication, so a solid understanding of vectors is the essential prerequisite.