Equation of a Line Through a Point and Parallel to a Vector in 3D

A line in three-dimensional space is uniquely determined by a point \(P(x_0, y_0, z_0)\) and a direction vector \(\vec{v} = \langle a, b, c \rangle\). Below are the three equivalent forms of its equation: Vector form, Parametric form, and Symmetric form. Use the interactive calculator to generate random examples, or input your own point and vector, then see a step‑by‑step solution.

Vector form: \( \mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{v} \)   →   \( \langle x,y,z \rangle = \langle x_0, y_0, z_0 \rangle + t \langle a,b,c \rangle,\; t\in\mathbb{R} \)

Parametric form: \( x = x_0 + a t,\; y = y_0 + b t,\; z = z_0 + c t,\; t\in\mathbb{R} \)

Symmetric form (if \(a,b,c \neq 0\)): \( \dfrac{x - x_0}{a} = \dfrac{y - y_0}{b} = \dfrac{z - z_0}{c} \). If any component of direction is zero, the symmetric form is adjusted accordingly (e.g., \(x = x_0\) constant).

3D Line Equation Solver

Point \( P(x_p, y_p, z_p) \) + Direction \( \vec{v} = \langle v_x, v_y, v_z \rangle \) → Vector, Parametric & Symmetric forms
Point Coordinates (through which line passes)
Direction Vector \( \vec{v} \) (parallel to line)
* Direction vector must have at least one non-zero component.
✨ Step-by-Step Solution ✨
STEP 1: Vector Form
The vector equation of a line is derived from the position vector \( \vec{r}_0 \) of the given point and the direction vector \( \vec{v} \). As the parameter \( t \) varies, the point \( \vec{r}(t) = \vec{r}_0 + t\vec{v} \) traces the entire line.
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STEP 2: Parametric Form
By equating the components of the vector equation, we obtain three scalar equations. Each coordinate is expressed linearly in terms of the parameter \( t \). This form is useful for plotting points.
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STEP 3: Symmetric Form
Eliminate the parameter \( t \) from the parametric equations. Solve for \( t \) from each non-zero component and set them equal. If a direction component is zero, the corresponding coordinate remains constant (e.g., \( x = x_0 \)).
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3D line through point P and parallel to vector v

Line through point \(P\) and parallel to \(\vec{v}\)

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