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Vectors

A vector quantity has both length (magnitude) and direction. The opposite is a scalar quantity, which only has magnitude. Vectors can be denoted by AB, a, or AB (with an arrow above the letters).
If a = then the vector will look as follows:

A line with an arrow from (0,0) to (3,2)

NB1: When writing vectors as one number above another in brackets, this is known as a column vector.
NB2: In textbooks and here, vectors are indicated by bold type. However, when you write them, you need to put a line underneath the vector to indicate it.

Multiplication by a Scalar

When multiplying a vector by a scalar (i.e. a number), multiply each component of the vector by that number.

Example

If a = , and b = 2a, sketch a and b.

If a = , then 2a =

A diagram showing a and 2a

Vector Manipulation

When adding two (or more) vectors, we add together the numbers in the same positions.

(4,1) + (1,-3) = (5,-2)

When multiplying a vector by a number (a "scalar"), we multiply each component (each bit) of the vector by the number.

If a=(4,1) then 2a = (8,2) and -a = (-4,-1)

The length or modulus of a vector a is denoted by |a|. If you look at a diagram of a vector, you should be able to see how to use Pythagoras's theorem to calculate the length of a vector:

The magnitude of (x,y) is the square root of x squared plus y squared

If a and b are parallel vectors (parallel means pointing in the same direction), then a will be a scalar multiple of b and vice-versa. So there will be a constant k with a = kb

Example

If a = and b = , find the magnitude of their resultant.

The resultant of two or more vectors is another word for their sum.
The resultant therefore is .
The magnitude of this is √(-3² + 4²) = √(9 + 16) = √(25) = 5

The addition and subtraction of vectors can be shown diagrammatically. To find a + b, draw a and then draw b at the end of a. The resultant is the line between the start of a and the end of b.
To find a - b, find -b (see above) and add this to a.

Example

A diagram showing (3, 2) + (4, -3)

Unit Vectors

A unit vector has a magnitude of 1. The unit vector in the direction of the x-axis is i and the unit vector in the direction of the y-axis is j. For example on a graph, 3i + 4j would be at (3 , 4). This method is another method of writing down vectors. It also makes adding and subtracting vectors easy: you just add the i terms together and add the j terms together.
For example: 3i + j plus 5i - 4j = 8i - 3j.