Vectors

This is a vector:

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.

Play with one here:

../geometry/images/geom-vector.js?mode=vector

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.

Subtracting

We can also subtract one vector from another:

first we reverse the direction of the vector we want to subtract,
then add them as usual:

vector subtract a-b = a + (-b)
a − b

Notation

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:

Calculations

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 a_x and a_y

(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)

When we break up a vector like that, each part is called a component:

../geometry/images/geom-vector.js?mode=xy

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:

|a|

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

||a||

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:

so c is a vector, it has magnitude and direction
but c is just a value, like 3 or 12.4

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 (5,2) by the scalar 3

a = 3(5,2) = (3×5,3×2) = (15,6)

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!

The scalar or Dot Product (the result is a scalar).
The vector or Cross Product (the result is a vector).

(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 a in Polar Coordinates		Vector a in Cartesian Coordinates

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 )