# Vector Equation — Line and Plane Forms, Examples

TL;DR

A vector equation describes a line or a plane using position vectors and a parameter, most often the line form \( \vec{r} = \vec{a} + \lambda \vec{b} \) — start at point \( \vec{a} \), then slide along direction \( \vec{b} \). This article covers the line form, the two-point form, the plane forms, what the parameter \( \lambda \) does, how to convert to Cartesian form, and six worked examples.

## What Is a Vector Equation?

A **vector equation** is an equation that uses vectors to describe a geometric object — usually a line or a plane in two or three dimensions. Instead of relating \( x \), \( y \), and \( z \) directly, it relates **position vectors** (arrows from the origin to a point) using a scalar **parameter**.

The most common vector equation is the equation of a line, \( \vec{r} = \vec{a} + \lambda \vec{b} \). Here \( \vec{r} \) is the position vector of _any_ point on the line, \( \vec{a} \) is the position vector of one known point, \( \vec{b} \) is a direction vector parallel to the line, and \( \lambda \) is the parameter you vary. The whole subject rests on combining vectors through addition and scalar multiplication.

## What Is the Vector Equation of a Line?

The **vector equation of a line** through a point with position vector \( \vec{a} \), parallel to direction vector \( \vec{b} \), is:

\[ \vec{r} = \vec{a} + \lambda \vec{b} \]

where \( \vec{r} \) is the position vector of a general point on the line and \( \lambda \) is a real-number parameter. Read it as a journey: travel from the origin out to point \( \vec{a} \), then move some multiple \( \lambda \) of the direction \( \vec{b} \).

### The two-point form

If you're given two points rather than a point and a direction, the direction _is_ the vector from one point to the other. For points with position vectors \( \vec{a} \) and \( \vec{b} \), the direction is \( \vec{b} - \vec{a} \), so:

\[ \vec{r} = \vec{a} + \lambda (\vec{b} - \vec{a}) \]

At \( \lambda = 0 \) you sit at \( \vec{a} \); at \( \lambda = 1 \) you land exactly on \( \vec{b} \). Any other value of \( \lambda \) places you somewhere along — or beyond — the segment joining them.

### What does the parameter \( \lambda \) actually do?

\( \lambda \) is the dial that picks out which point on the line you mean. Each value of \( \lambda \) names exactly one point:

- \( \lambda = 0 \) gives the base point \( \vec{a} \).
- \( \lambda = 1 \) moves you one full copy of \( \vec{b} \) along the line.
- \( \lambda = 2 \) moves you two copies along — twice as far.
- \( \lambda = -1 \) moves you one copy in the _opposite_ direction.

As \( \lambda \) runs over every real number, \( \vec{r} \) sweeps out the entire infinite line. This is exactly the parametric idea: one variable, the whole curve.

## What Is the Vector Equation of a Plane?

A plane needs more pinning down than a line, and there are three standard vector equations for it depending on what you're given.

- **Normal form** — a plane at perpendicular distance \( d \) from the origin with unit normal \( \hat{n} \):  \( \vec{r} \cdot \hat{n} = d \).
- **Point-normal form** — a plane through point \( \vec{a} \) with normal vector \( \vec{N} \): \( (\vec{r} - \vec{a}) \cdot \vec{N} = 0 \).
- **Three-point form** — a plane through three non-collinear points \( \vec{a} \), \( \vec{b} \), \( \vec{c} \): \( (\vec{r} - \vec{a}) \cdot \big[(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})\big] = 0 \).

## How Do You Convert a Vector Equation to Cartesian Form?

The Cartesian form drops out when you compare components and eliminate \( \lambda \). Take a line \( \vec{r} = \vec{a} + \lambda \vec{b} \) with \( \vec{a} = (x_1, y_1, z_1) \) and \( \vec{b} = (b_1, b_2, b_3) \). Writing \( \vec{r} = (x, y, z) \) component by component gives:

\[ x = x_1 + \lambda b_1, \quad y = y_1 + \lambda b_2, \quad z = z_1 + \lambda b_3 \]

Solve each for \( \lambda \) and set them equal:

\[ \frac{x - x_1}{b_1} = \frac{y - y_1}{b_2} = \frac{z - z_1}{b_3} \]

That symmetric form is the Cartesian equation of the line. The vector form and the Cartesian form describe the same line — one keeps the parameter, the other eliminates it.

## Examples of Vector Equation

### Example 1

**Find the vector equation of the line through the point \( (1, 2, 3) \) parallel to \( \vec{b} = (2, -1, 4) \).**

The base point gives \( \vec{a} = (1, 2, 3) \) and the direction is \( \vec{b} = (2, -1, 4) \). Drop them into \( \vec{r} = \vec{a} + \lambda \vec{b} \):

\[ \vec{r} = (1, 2, 3) + \lambda(2, -1, 4) \]

**Final answer:** \( \vec{r} = (1, 2, 3) + \lambda(2, -1, 4) \).

### Example 2

**Find the vector equation of the line through A\( (1, 0, 2) \) and B\( (4, 3, 2) \).**

_Correct._ The direction between two points is \( \vec{b} - \vec{a} \):

\[ \vec{r} = (1, 0, 2) + \lambda((4, 3, 2) - (1, 0, 2)) \]

**Final answer:** \( \vec{r} = (1, 0, 2) + \lambda(3, 3, 0) \).

### Example 3

**Find the vector equation of the line through \( (0, 1, -1) \) and \( (2, 1, 3) \).**

Direction: \( (2, 1, 3) - (0, 1, -1) = (2, 0, 4) \)

Using \( \vec{a} = (0, 1, -1) \):

\[ \vec{r} = (0, 1, -1) + \lambda(2, 0, 4) \]

**Final answer:** \( \vec{r} = (0, 1, -1) + \lambda(2, 0, 4) \).

### Example 4

**Convert \( \vec{r} = (1, 2, 3) + \lambda(2, -1, 4) \) to Cartesian form.**

Read off the base point and direction, then build the symmetric form:

\[ \frac{x - 1}{2} = \frac{y - 2}{-1} = \frac{z - 3}{4} \]

**Final answer:** \( \frac{x - 1}{2} = \frac{y - 2}{-1} = \frac{z - 3}{4} \).

### Example 5

**Find the vector equation of the plane through \( (1, 0, 0) \) with normal \( \vec{N} = (2, 3, 1) \).**

Using point-normal form:

\[ (\vec{r} - (1, 0, 0)) \cdot (2, 3, 1) = 0 \]

**Final answer:** \( 2(x - 1) + 3y + z = 0 \), i.e. \( 2x + 3y + z = 2 \).

### Example 6

**Find the vector equation of the plane through three points A\( (1, 0, 0) \), B\( (0, 1, 0) \), C\( (0, 0, 1) \).**

Build two in-plane directions. Their cross product gives the normal:

\[ (\vec{b} - \vec{a}) \times (\vec{c} - \vec{a}) = (1, 1, 1) \]

Then:

\[ (\vec{r} - \vec{a}) \cdot (1, 1, 1) = 0 \]

**Final answer:** \( x + y + z = 1 \).

## Why Vector Equations Are Worth Learning

Vector equations were the natural language once mathematicians started treating direction as a first-class object.

Where the position-plus-direction idea pays off:
- **Computer graphics and games.** A ray from the camera through a pixel is exactly \( \vec{r} = \vec{a} + \lambda \vec{b} \).
- **Robotics and flight paths.** A drone's trajectory is a base position plus a velocity direction scaled by time; \( \lambda \) becomes the clock.
- **3D modelling and CAD.** Surfaces are planes written in normal form; intersecting them is solving two plane equations.
- **Physics.** The path of a particle moving at constant velocity is a vector equation of a line.

## Key Takeaways
- A **vector equation** describes a line or plane using position vectors and a parameter.
- The line form is \( \vec{r} = \vec{a} + \lambda \vec{b} \): start at point \( \vec{a} \), slide along direction \( \vec{b} \).
- For a line through two points, the direction is \( \vec{b} - \vec{a} \).
- The parameter \( \lambda \) names each point.
- A plane uses the point-normal form \( (\vec{r} - \vec{a}) \cdot \vec{N} = 0 \).
- The same line has infinitely many vector equations.