Vectors

This is a vector:

A vector has magnitude (how long it is) and direction:

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

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

And it doesn't matter which order you add them, you 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.

Subtracting

You can also subtract one vector from another:

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

Other Notation

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

Calculations

Now ... how do we do the calculations?

The most common way is to break up a vector into x and y pieces, like this:

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

Adding Vectors

And here is how to add two vectors after breaking them into x and 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

Remember: to subtract, first reverse the vector you 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||

You can use Pythagoras' theorem to calculate it:

|a| = √( x2 + y2 )

A vector with magnitude 1 is called a Unit Vector.

Vector vs Scalar

When using vectors we call an ordinary number a "scalar".

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

A vector is often written in bold,

so c is a vector, and 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 you multiply a vector by a scalar it is called "scaling" a vector, because you 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 you multiply two vectors together? There is more than one way!

(Read those pages for more details.)

 

More Than 2 Dimensions

The vectors we have been looking at have been 2 dimensional, but vectors work perfectly well in 3 or more dimensions:

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: 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)

 

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

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

 

Magnitude and Direction

You 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 )

 

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:

Now, convert from polar to Cartesian (to 2 decimals):

Sam's Vector:

  • x = r × cos( θ ) = 200 × cos(60°) = 200 × 0.5 = 100
  • y = r × sin( θ ) = 200 × sin(60°) = 200 × 0.8660 = 173.21

Alex's Vector:

  • x = r × cos( θ ) = 120 × cos(-45°) = 120 × 0.7071 = 84.85
  • y = r × sin( θ ) = 120 × sin(-45°) = 120 × -0.7071 = -84.85

Now we have:

Now it is easy to add them:

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

We can convert that to polar for a final answer:

  • r = √ ( x2 + y2 ) = √ ( 184.852 + 88.362 ) = 204.88
  • θ = tan-1 ( y / x ) = tan-1 ( 88.36 / 184.85 ) = 25.5°
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!