This is a vector:


A vector has magnitude (size) and direction:

vector magnitude and direction

The length of the line shows its magnitude and the arrowhead points in the direction.

We can add two vectors by joining them head-to-tail:

vector add a+b

And it doesn't matter which order we add them, we get the same result:

vector add b+a

Example: A plane is flying along, pointing North, but there is a wind coming from the North-West.

vector airplane, propellor and wind

The two vectors (the velocity caused by the propeller, and the velocity of the wind) result in a slightly slower ground speed heading a little East of North.

If you watched the plane from the ground it would seem to be slipping sideways a little.

vector airplane ahead and slightly sideways

Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seem to fly sideways. Vectors help explain that.

Velocity, acceleration, force and many other things are vectors.


We can also subtract one vector from another:

vector subtract a-b = a + (-b)


A vector is often written in bold, like a or b.

A vector can also be written as the letters
of its head and tail with an arrow above it, like this:
  vector notation a=AB, head, tail


Now ... how do we do the calculations?

The most common way is to first break up vectors into x and y parts, like this:

vector xy components

The vector a is broken up into
the two vectors ax and ay

(We see later how to do this.)

Adding Vectors

We can then add vectors by adding the x parts and adding the y parts:

vector add example

The vector (8,13) and the vector (26,7) add up to the vector (34,20)

Example: add the vectors a = (8,13) and b = (26,7)

c = a + b

c = (8,13) + (26,7) = (8+26,13+7) = (34,20)

Subtracting Vectors

To subtract, first reverse the vector we want to subtract, then add.

Example: subtract k = (4,5) from v = (12,2)

a = v + −k

a = (12,2) + −(4,5) = (12,2) + (−4,−5) = (12−4,2−5) = (8,−3)

Magnitude of a Vector

The magnitude of a vector is shown by two vertical bars on either side of the vector:


OR it can be written with double vertical bars (so as not to confuse it with absolute value):


We use Pythagoras' theorem to calculate it:

|a| = √( x2 + y2 )

Example: what is the magnitude of the vector b = (6,8) ?

|b| = √( 62 + 82 ) = √( 36+64 ) = √100 = 10

A vector with magnitude 1 is called a Unit Vector.

Vector vs Scalar

A scalar has magnitude (size) only.

Scalar: just a number (like 7 or −0.32) ... definitely not a vector.

A vector has magnitude and direction, and is often written in bold, so we know it is not a scalar:

Example: kb is actually the scalar k times the vector b.

Multiplying a Vector by a Scalar

When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is.

Example: multiply the vector m = (7,3) by the scalar 3

vector scaling   a = 3m = (3×7,3×3) = (21,9)

It still points in the same direction, but is 3 times longer

(And now you know why numbers are called "scalars", because they "scale" the vector up or down.)


Multiplying a Vector by a Vector (Dot Product and Cross Product)

dot product magnitude and angle

How do we multiply two vectors together? There is more than one way!

(Read those pages for more details.)


More Than 2 Dimensions

Vectors also work perfectly well in 3 or more dimensions:

vector in 3d
The vector (1,4,5)

Example: add the vectors a = (3,7,4) and b = (2,9,11)

c = a + b

c = (3,7,4) + (2,9,11) = (3+2,7+9,4+11) = (5,16,15)

Example: what is the magnitude of the vector w = (1,−2,3) ?

|w| = √( 12 + (−2)2 + 32 ) = √( 1+4+9 ) = √14

Here is an example with 4 dimensions (but it is hard to draw!):

Example: subtract (1,2,3,4) from (3,3,3,3)

(3,3,3,3) + −(1,2,3,4)
= (3,3,3,3) + (−1,−2,−3,−4)
= (3−1,3−2,3−3,3−4)
= (2,1,0,−1)


Magnitude and Direction

We may know a vector's magnitude and direction, but want its x and y lengths (or vice versa):

vector polar <=> vector cartesian
Vector a in Polar
  Vector a in Cartesian

You can read how to convert them at Polar and Cartesian Coordinates, but here is a quick summary:

From Polar Coordinates (r,θ)
to Cartesian Coordinates (x,y)
  From Cartesian Coordinates (x,y)
to Polar Coordinates (r,θ)
  • x = r × cos( θ )
  • y = r × sin( θ )
  • r = √ ( x2 + y2 )
  • θ = tan-1 ( y / x )



vector example two people pull

An Example

Sam and Alex are pulling a box.

What is the combined force, and its direction?


Let us add the two vectors head to tail:

vectors: angles and magnitudes

First convert from polar to Cartesian (to 2 decimals):

Sam's Vector:

Alex's Vector:

Now we have:

vectors: components

Add them:

(100, 173.21) + (84.85, −84.85) = (184.85, 88.36)

That answer is valid, but let's convert back to polar as the question was in polar:

And we have this (rounded) result:
vector result

And it looks like this for Sam and Alex:
vector combined pull force

They might get a better result if they were shoulder-to-shoulder!