VECTORS

A vector is a pair of points e.g. .

You can think of it as a journey, starting at a and ending at b.

NOTE: is usually written as . A single letter vector always starts at the point o.

Equality

Two vectors are equal if they have the same length and the same direction.

Example

Write down four pairs of equal vectors in the following diagram. Note: or || pq and op || rq.

Answer.

Addition of Vectors

THE TRIANGLE LAW

If one vector starts where another ends we can add them according to the triangle law:

THE PARALLELOGRAM LAW

If two vectors start at the same point, we use the parallelogram law.

To add two vectors, and as in diagram, complete the parallelogram. The sum of the two vectors is given by the blue diagonal.

Negative Vectors.

The Universal Law

This law is used constantly. It is essential for converting a two letter vector to a "single" letter vector.
(Remember that .)

Multiplying a Vector by a Number

Multiplying a vector by a postive number, e.g. 3, increases its length by a factor of, e.g. 3..

Multiplying a vector by a negative number, e.g. -7, increases its length by a factor of, e.g. 7, and reverses its direction.

i-j Coordinates.

is a vector one step right. is a vector one step up.

Modulus of a Vector

The modulus of , written , is the distance from o to a.

From Pythagoras' theorem

Dot Product

Dot product is used to find the angle between to vectors or to show whether vectors are parallel or perpendicular.

DEFINITION     

THEOREM       

THEOREM

Unit Vectors

If then the unit vector in the direction of is given by

The Related Perpendicular Vector

If , then the related perpendicular vector (r-perp for short) is given by

(NOTE: minus the middle then swap)