This is a vector:

A vector has magnitude (size) 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:

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

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

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.

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:



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:


Now ... how do we do the calculations?

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

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:

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

  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)

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:

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 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 )


An Example

Sam and Alex are pulling a box.

  • Sam pulls with 200 Newtons of force at 60°
  • Alex pulls with 120 Newtons of force at 45° as shown

What is the combined force, and its direction?

Let us add the two vectors head to tail:

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

Sam's Vector:

Alex's Vector:

Now we have:

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:

And it looks like this for Sam and Alex:

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