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Vector Equation of a Line

A 3D line through point r₀ with direction v: r = r₀ + tv. Parametric: x = x₀+at, y = y₀+bt, z = z₀+ct. Symmetric: (x−x₀)/a = (y−y₀)/b = (z−z₀)/c. Used in ray tracing and 3D geometry.

Concept Fundamentals
r = r₀ + tv
Vector
x = x₀+at, y = y₀+bt, z = z₀+ct
Parametric
(x−x₀)/a = (y−y₀)/b = (z−z₀)/c
Symmetric
v = (a,b,c)
Direction

Did our AI summary help? Let us know.

t is the parameter along the line. Direction vector v can be any non-zero multiple. Symmetric form fails when a, b, or c is 0.

Key quantities
r = r₀ + tv
Vector
Key relation
x = x₀+at, y = y₀+bt, z = z₀+ct
Parametric
Key relation
(x−x₀)/a = (y−y₀)/b = (z−z₀)/c
Symmetric
Key relation
v = (a,b,c)
Direction
Key relation

Ready to run the numbers?

Why: Vector line equations are used in ray tracing, collision detection, and 3D graphics. Parametric form is ideal for computing points on the line; symmetric form for intersection tests.

How: Given point (x₀,y₀,z₀) and direction (a,b,c): vector form r = r₀ + tv. Parametric: x = x₀+at, etc. Symmetric when a,b,c ≠ 0: (x−x₀)/a = (y−y₀)/b = (z−z₀)/c.

t is the parameter along the line.Direction vector v can be any non-zero multiple.
Sources:Khan Academy — 3D LinesWolfram MathWorld — Line

Run the calculator when you are ready.

Find Line EquationEnter point and direction

Sample Examples

Input

Point on Line (r₀)

Direction Vector (v)

Results

Vector: r = (1, 2, 3) + t(1, 1, 1)
Parametric: x = 1 + 1t, y = 2 + 1t, z = 3 + 1t
Symmetric: (x-1)/1 = (y-2)/1 = (z-3)/1

Step-by-Step

Point r₀ = (1, 2, 3)

Direction v = (1, 1, 1)

Vector form: r = r₀ + tv

Parametric: x = x₀ + at, y = y₀ + bt, z = z₀ + ct

Symmetric: (x-1)/1 = (y-2)/1 = (z-3)/1

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

→

r = r₀ + tv (vector form).

— Line Equation

t

Parametric: x = x₀+at, y = y₀+bt, z = z₀+ct.

— Parametric

Key Takeaways

  • Vector form: vecr=vecr0+tvecv\\vec{r} = \\vec{r}_0 + t\\vec{v}, where r₀ is a point and v is the direction vector.
  • Parametric: x=x0+at,quady=y0+bt,quadz=z0+ctx = x_0 + at,\\quad y = y_0 + bt,\\quad z = z_0 + ct.
  • Symmetric: fracx−x0a=fracy−y0b=fracz−z0c\\frac{x-x_0}{a} = \\frac{y-y_0}{b} = \\frac{z-z_0}{c} (when a,b,c ≠ 0).
  • Direction vector must be non-zero.
  • Parallel lines have proportional direction vectors.

Did You Know?

Ray tracing uses line equations to find intersections with surfaces.
Robotics uses parametric lines for motion planning.
When one direction component is zero, symmetric form has fewer terms.
Two skew lines in 3D never intersect and aren't parallel.
Physics uses these for particle trajectories.
CAD software represents edges as parametric lines.

Understanding

A line in 3D is determined by a point and a direction. The parameter t sweeps through all real numbers.

vecr=vecr0+tvecv\\vec{r} = \\vec{r}_0 + t\\vec{v}
x=x0+at,quady=y0+bt,quadz=z0+ctx = x_0 + at,\\quad y = y_0 + bt,\\quad z = z_0 + ct
fracx−x0a=fracy−y0b=fracz−z0c\\frac{x-x_0}{a} = \\frac{y-y_0}{b} = \\frac{z-z_0}{c}

Expert Tips

Normalize the direction vector for unit-speed parametrization.
Check a,b,c ≠ 0 before writing symmetric form.
Use t=0 for the given point, t=1 for point + direction.
Dot product of direction with plane normal gives line-plane angle.

FAQ

Q: What if direction vector is zero?
A: Not a valid line; need non-zero direction.
Q: Can direction have negative components?
A: Yes; it only affects which way t increases.
Q: How to find line through two points?
A: Use one point and direction = P₂ - P₁.
Q: When is symmetric form undefined?
A: When any of a,b,c is zero; omit that term.
Q: What is the difference from 2D line?
A: 3D adds z component; same parametric idea.
Q: How to check if point is on line?
A: Find t such that point = r₀ + tv; solve for t.
Q: Applications in computer graphics?
A: Ray casting, clipping, path animation.

How to Use

  1. Enter point on line (x₀, y₀, z₀).
  2. Enter direction vector (a, b, c).
  3. View vector, parametric, and symmetric forms.

Disclaimer

Direction vector must be non-zero. Symmetric form may omit terms when components are zero.

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