Equation of Line 3D Calculator
Verified Calculation Engine
The online Equation of Line 3D Calculator helps you calculate instantly and solve problems related to Vectors 3d Geometry. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Math Numbers calculations.
Finds equation of line in 3D: vector form r⃗ = a⃗ + λb⃗, parametric form (x, y, z) = (x₁, y₁, z₁) + λ(a, b, c), or symmetric form (x-x₁)/a = (y-y₁)/b = (z-z₁)/c.
Line through point (x₁, y₁, z₁) with direction (a, b, c):
Vector form: r⃗ = (x₁, y₁, z₁) + λ(a, b, c)
Parametric: x = x₁ + λa, y = y₁ + λb, z = z₁ + λc
Symmetric: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
Line equation in 3D using point and direction
Inputs
Results
Worked Examples
Example 1: Basic
Find equation of line through (1, 2, 3) with direction (2, 3, 4)
- point: (1, 2, 3)
- direction: (2, 3, 4)
- vectorForm: r⃗ = (1, 2, 3) + λ(2, 3, 4)
- parametricForm: x = 1 + 2λ, y = 2 + 3λ, z = 3 + 4λ
- symmetricForm: (x-1)/2 = (y-2)/3 = (z-3)/4
Example 2: Board Level
Find equation of line through (0, 0, 0) with direction (1, 1, 1)
- point: (0, 0, 0)
- direction: (1, 1, 1)
- vectorForm: r⃗ = λ(1, 1, 1)
- parametricForm: x = λ, y = λ, z = λ
- symmetricForm: x/1 = y/1 = z/1
About this calculator
Overview
Equation of Line 3D Calculator evaluates Equation of Line 3D for the math / vectors-3d-geometry library. Primary input cue: Point on Line [required]: Point (x₁, y₁, z₁) on line. Method anchor: Line through point (x₁, y₁, z₁) with direction (a, b, c): Vector form: r⃗ = (x₁, y₁, z₁) + λ(a, b, c) Parametric: x = x₁ + λa, y = y₁ + λb, z = z₁ + λc Symmetri.
When to use
Reach for Equation of Line 3D Calculator to compare scenarios by changing one input at a time for equation of line 3d.
Inputs explained
- Point on Line [required]: Point (x₁, y₁, z₁) on line
- Direction Vector or Ratios [required]: Direction vector (a, b, c) or direction ratios
Formula / method
Line through point (x₁, y₁, z₁) with direction (a, b, c): Vector form: r⃗ = (x₁, y₁, z₁) + λ(a, b, c) Parametric: x = x₁ + λa, y = y₁ + λb, z = z₁ + λc Symmetric: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c. Line equation in 3D using point and direction (calculator id: equation-of-line-3d-calculator).
Worked example
Worked values: point = (1, 2, 3), direction = (2, 3, 4) → vectorForm = r⃗ = (1, 2, 3) + λ(2, 3, 4), parametricForm = x = 1 + 2λ, y = 2 + 3λ, z = 3 + 4λ, symmetricForm = (x-1)/2 = (y-2)/3 = (z-3)/4
Interpreting results
Read the primary output as equation of line 3d under the method on this page.
Assumptions
- Units (when shown) must be consistent across inputs for equation of line 3d. [equation-of-line-3d-calculator]
- Empty required inputs are not silently replaced with zeros for equation of line 3d.
- Real-number arithmetic is used unless equation of line 3d explicitly needs integers.
Limitations
- This page does not provide a full textbook proof for every identity related to equation of line 3d.
- Equation of Line 3D Calculator cannot invent missing inputs; incomplete forms stop short of a forged equation of line 3d value.
- Complex/branch cuts and undefined points are not always interactively annotated for equation of line 3d.
Important note
Educational mathematics tool. Check edge cases and domain restrictions when results look unexpected.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
Line through point (x₁, y₁, z₁) with direction (a, b, c): Vector form: r⃗ = (x₁, y₁, z₁) + λ(a, b, c) Parametric: x = x₁ + λa, y = y₁ + λb, z = z₁ + λc Symmetric: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c. Line equation in 3D using point and direction (calculator id: equation-of-line-3d-calculator).
Example Calculation
Worked values: point = (1, 2, 3), direction = (2, 3, 4) → vectorForm = r⃗ = (1, 2, 3) + λ(2, 3, 4), parametricForm = x = 1 + 2λ, y = 2 + 3λ, z = 3 + 4λ, symmetricForm = (x-1)/2 = (y-2)/3 = (z-3)/4
Frequently Asked Questions
What is Equation of Line 3D Calculator?
- Equation of Line 3D Calculator evaluates Equation of Line 3D for the math / vectors-3d-geometry library. Primary input cue: Point on Line [required]: Point (x₁, y₁, z₁) on line. Method anchor: Line through point (x₁, y₁, z₁) with direction (a, b, c): Vector form: r⃗ = (x₁, y₁, z₁) + λ(a, b, c) Parametric: x = x₁ + λa, y = y₁ + λb, z = z₁ + λc Symmetri.
How does Equation of Line 3D Calculator work?
- Line through point (x₁, y₁, z₁) with direction (a, b, c): Vector form: r⃗ = (x₁, y₁, z₁) + λ(a, b, c) Parametric: x = x₁ + λa, y = y₁ + λb, z = z₁ + λc Symmetric: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c. Line equation in 3D using point and direction (calculator id: equation-of-line-3d-calculator).
Why use this Math Numbers calculator?
- Reach for Equation of Line 3D Calculator to compare scenarios by changing one input at a time for equation of line 3d.