# Vector Form - Definition, Formula, and Examples

## What Is Vector Form?

**Vector form** is a way of writing the equation of a line (or a plane) using vectors instead of plain x and y coordinates. A **vector** is a quantity with both size and direction, drawn as an arrow; if you need the basics first, see [what is a vector](/content/math/terms/vector/index.html) and the fuller treatment of [vectors](/content/math/geometry/vectors/index.html).

The core idea is simple. To pin down a straight line, you only need two things:

- **One point on the line**, given by its **position vector** \(\vec{a}\) (the arrow from the origin to that point), and
- **The direction the line runs**, given by a **direction vector** \(\vec{b}\).

From there, every point on the line is reachable by starting at \(\vec{a}\) and travelling some multiple of \(\vec{b}\). That multiple is the scalar \(\lambda\) (lambda), and it can be any real number - positive, negative, or zero.

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

As \(\lambda\) changes, \(\vec{r}\) sweeps out every point on the line. Set \(\lambda = 0\) and you are back at \(\vec{a}\); set \(\lambda = 1\) and you have moved one full direction-vector along; make \(\lambda\) negative and you travel backwards.

### The two standard forms

There are two ways a line is usually written in vector form, and both are the same formula wearing different clothes.

**Through a point, parallel to a direction:**

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

where \(\vec{a}\) is the position vector of a known point and \(\vec{b}\) is the direction vector.

**Through two known points** with position vectors \(\vec{a}\) and \(\vec{c}\):

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

Here the direction vector is built by subtracting the two points, \(\vec{c} - \vec{a}\), which is the arrow pointing from the first point to the second. This is why the vector concepts, [parallel vectors](/content/math/geometry/parallel-vectors/index.html) and the [position vector](/content/math/geometry/position-vector/index.html), are essential in vector form: the direction \(\vec{b}\) is direction, and \(\vec{a}\) is a position.

### How vector form links to parametric and Cartesian forms

Vector form is not a rival to the equations you already know - it is their parent. Write \(\vec{r} = (x,y,z)\), \(\vec{a} = (a_1,a_2,a_3)\), and \(\vec{b} = (b_1,b_2,b_3)\), then read off one coordinate equation per axis:

\[ x = a_1 + \lambda b_1 \]
\[ y = a_2 + \lambda b_2 \]
\[ z = a_3 + \lambda b_3 \]

Those three lines are the **parametric form** - the same line, written coordinate by coordinate. Eliminate \(\lambda\) from them and you get the **Cartesian (symmetric) form**. All three describe the identical set of points; vector form is just the most compact.

| Form      | Looks like                                           | Best for                           |
|-----------|-----------------------------------------------------|------------------------------------|
| Vector    | \(\vec{r} = \vec{a} + \lambda \vec{b}\)               | compact, works in 3D              |
| Parametric| \(x = a_1 + \lambda b_1\), etc.                 | plugging in a parameter            |
| Cartesian | \(\frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}\) | no parameter, direct relation      |

## Examples Of Vector Form

These build from writing a line up to finding where a point sits on it. One of them shows a wrong turn worth walking through.

### Example 1

**Write the vector equation of a line passing through the point (2,3) with direction vector \(\vec{b}=4\hat{i}+5\hat{j}\).**

The position vector of the point is \(\vec{a}=2\hat{i}+3\hat{j}\). Slot both into the standard form.

\[ \vec{r} = \vec{a} + \lambda \vec{b} \]
\[ \vec{r} = (2\hat{i}+3\hat{j}) + \lambda(4\hat{i}+5\hat{j}) \]

That is the vector equation of the line.

### Example 2

**Write the vector equation of the line through the two points A(1,2,3) and C(4,5,6).**

The tempting move is to add the two position vectors to get a direction: \(\vec{a} + \vec{c} = (1+4,2+5,3+6)=(5,7,9)\), then write \(\vec{r} = \vec{a} + \lambda(5,7,9)\).

That is wrong, and here is the tell: adding two positions gives a point that is roughly "between and beyond" the two, not the direction _from one to the other_. A direction must describe travel, so it comes from **subtraction**, not addition.

The correct direction vector is \(\vec{c} - \vec{a}\):

\[ \vec{c} - \vec{a} = (4 - 1, 5 - 2, 6 - 3) = (3, 3, 3) \]

So the line is

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

Check it: \(\lambda=0\) gives A(1,2,3) and \(\lambda=1\) gives (4,5,6)=C. Both endpoints land, so the direction was right.

### Example 3

**Find the point on the line \(\vec{r} = (1,0,2)+\lambda(2,1,3)\) when \(\lambda=2\).**

Substitute \(\lambda=2\) and compute each coordinate:

\[ x = 1 + 2(2) = 5 \]
\[ y = 0 + 1(2) = 2 \]
\[ z = 2 + 3(2) = 8 \]

The point is (5,2,8).

### Example 4

**Convert the vector equation \(\vec{r} = (3,-1)+\lambda(2,4)\) into parametric form.**

Read off one equation per coordinate:

\[ x = 3 + 2\lambda \]
\[ y = -1 + 4\lambda \]

These are the parametric form of the same line.

### Example 5

**Is the point (7,7) on the line \(\vec{r} = (1,3)+\lambda(3,2)\)?**

Write the parametric equations and solve for \(\lambda\) from one coordinate:

For x:
\[ 7 = 1 + 3\lambda \Rightarrow \lambda = 2 \]

Then check y:
\[ 3 + 2(2) = 7 \]

Both coordinates satisfy, so yes, (7,7) lies on the line.

### Example 6

**Give the direction vector of the line \(\vec{r}=(0,4,-2)+\lambda(5,-3,1)\), and state whether (10,-2,0) is on it.**

The direction vector is \(\vec{b}=(5,-3,1)\). Test the point coordinate by coordinate:

- For x:
  \[ 10 = 0 + 5\lambda \Rightarrow \lambda = 2 \]
- For y:
  \[ 4 + (-3)(2) = -2 \checkmark \]
- For z:
  \[ -2 + 1(2) = 0 \checkmark \]

All three match at \(\lambda = 2\), so (10,-2,0) is on the line.

## Why Vector Form Matters - "The Only Line Equation That Survives In 3D"

Vector form exists because the school-favourite \(y = mx + c\) quietly fails the moment you leave a flat page. In three dimensions, there is no single "slope" for a line, and a vertical line has no slope at all. Vector form sidesteps both problems.

- **It works in any number of dimensions.** A point plus a direction defines a line whether you are in 2D, 3D, or higher. There is nothing to break because \(\vec{r} = \vec{a} + \lambda \vec{b}\) never mentions slope.
- **It matches how motion actually happens.** A moving object _is_ a starting position plus time along a velocity direction, which is precisely \(\vec{a} + \lambda \vec{b}\) where \(\lambda\) stands in for time.
- **Where the maths is going.** Once a line is \(\vec{a} + \lambda \vec{b}\), you can ask whether two lines run in parallel directions (their direction vectors are multiples), whether they meet, and what angle they make, using the [scalar product](/content/math/geometry/scalar-product/index.html). The same \(\vec{r} = \vec{a} + \lambda \vec{b}\) idea extends to planes and underpins the wider algebra of the [multiplication of vectors](/content/math/algebra/multiplication-of-vectors/index.html).

## Mistakes To Watch For With Vector Form

### Mistake 1: Adding two points instead of subtracting for the direction

**Where it slips in:** finding the vector equation of a line through two given points.  
**Don't do this:** use \(\vec{a} + \vec{c}\) as the direction vector.  
**The correct way:** the direction from point A to point C is \(\vec{c} - \vec{a}\), always a subtraction.

### Mistake 2: Treating the position vector as the direction (or vice versa)

**Where it slips in:** reading a line written as \(\vec{r} = \vec{a} + \lambda \vec{b}\).  
**Don't do this:** call \(\vec{a}\) the direction and \(\vec{b}\) the point.
**The correct way:** \(\vec{a}\) is the _position_ of a known point; \(\vec{b}\) is the _direction_.

### Mistake 3: Thinking different equations mean different lines

**Where it slips in:** comparing your answer to a textbook's when both use different points or a scaled direction.  
**Don't do this:** assume \(\vec{r}=(1,2,3)+\lambda(3,3,3)\) and \(\vec{r}=(4,5,6)+\mu(1,1,1)\) are different lines.  
**The correct way:** a line's vector equation is not unique - any point on the line works as \(\vec{a}\), and any scalar multiple of \(\vec{b}\) works as the direction.

## Key Takeaways

- **Vector form** writes a line as \(\vec{r} = \vec{a} + \lambda \vec{b}\): a **position vector** \(\vec{a}\) plus a scalar \(\lambda\) times a **direction vector** \(\vec{b}\).
- Through two points, the direction is \(\vec{c} - \vec{a}\), a **subtraction**, never an addition.
- Splitting into coordinates gives **parametric form**; eliminating \(\lambda\) gives **Cartesian form** - same line, three notations.
- Vector form works in **any dimension** and is the standard for 3D lines and planes.
- A line's vector equation is **not unique** - many points and scaled directions describe the same line.