Vectors

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.

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:

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:

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)

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:

|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 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!

• 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:

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 )

 

 

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:

• 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:

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:

• 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!