Data Science : What are vectors, Maths!


This is a vector:


A vector has a 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 seems to fly sideways. Vectors help explain that.

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


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


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

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.

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

scalar has a magnitude (size) only.

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

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.

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