Equation of a Line Through Two Points in 3D

Find the vector, parametric, and symmetric equations of a line in space passing through two points. Enter coordinates below or generate random examples. Step-by-step solutions provided.

Line through \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \)

🔹 Direction vector: \( \vec{v} = \langle x_2-x_1,\; y_2-y_1,\; z_2-z_1 \rangle \)
🔹 Vector form: \( \mathbf{r}(t) = \langle x_1,y_1,z_1 \rangle + t \vec{v} \)
🔹 Parametric: \( x = x_1 + v_x t,\; y = y_1 + v_y t,\; z = z_1 + v_z t \)
🔹 Symmetric: \( \frac{x-x_1}{v_x} = \frac{y-y_1}{v_y} = \frac{z-z_1}{v_z} \) (if components non-zero)

Note: If a direction component is zero, symmetric form adjusts (e.g., \(x = x_1\) constant).

3D Line from Two Points

Vector · Parametric · Symmetric — step by step
Point P (x₁, y₁, z₁)
Point Q (x₂, y₂, z₂)

✨ Step-by-Step Solution

STEP 1: Direction Vector \( \vec{PQ} \)
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STEP 2: Vector Equation
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STEP 3: Parametric Equations
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STEP 4: Symmetric Form
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t ∈ ℝ — any real number generates points on the line. If a direction component = 0, symmetric equation uses constant coordinate.
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