VECTORS AND TRANSLATIONS

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:

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 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:

The vector (8, 13) and the vector (26, 7) add up to the vector (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.

The 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 )

A vector with magnitude 1 is called a Unit Vector.

Vector vs Scalar

A scalar has magnitude (size) only.

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

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.)

Here is an example with 4 dimensions (but it is hard to draw!):

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

TRANSLATIONS