Higher Mathematics Unit 3.1 Vectors

1. Introduction A vector is a quantity with both magnitude and direction. It can be represented using a direct line segment This vector is named or u A B

2. Vectors in 3 - Dimensions

-2 4 3

3 2 0

3. Finding the components of a Vector from Coordinates

P (1, 2) Q (6, 3)

S (-2, 1) T (5, 3)

A (-2, -1) B (4, 1)

4. Magnitude

4242 (-3)

5. Adding Vectors

A B C D Add vectors “ Nose-to-tail”

u u + v Add vectors “ Nose-to-tail” v

u A B -u A B is the negative of

u Add the negative of the vector “ Nose-to-tail” -v v u + -v u - v -2 -4

v The Zero Vector -v Back to the start. Gone nowhere

7. Multiplication by a Scalar

v 2v 2v has TWICE the MAGNITUDE of v, but v and 2v have the SAME DIRECTION. i.e. They are PARALLEL

8. Position Vectors

p P (4, 2) The position vector of a point P is the vector from the origin O, to P. The position vector is denoted by 4 2 If P has coordinates (x, y, z) then the components of the position vector of P are

9. Collinear points

A B E D C NOT collinear Collinear then the vectors are parallel and have a point in common - namely B -, this makes them collinear

10. Dividing lines in given ratios “Section Formula”

Give up John, they are getting bored!!

11. Unit Vectors

A unit vector is any vector whose length (magnitude) is one The vector is a unit vector since

There are three special unit vectors:

All vectors can be represented using a sum of these unit vectors

12. Scalar Product

The scalar product (or “dot” product) is a kind of vector “multiplication”. It is quite different from any kind of multiplication we’ve met before. The scalar product of the vectors and is defined as: or where  is the angle between the vectors, pointing out from the vertex

Calculating the angle between two vectors We have already seen that Rearranging gives And hence we can find the angle between two vectors

Some important results using the scalar product 3. Perpendicular vectors :  Provided and are non zero then if then so ie and are perpendiculiar 1.The scalar product is a number not a vector 2. If either or then 4.