Vectors Vectors are represented by a directed line segment its length representing the magnitude and an arrow indicating the direction A B or u u This vector is named

v === -v = w == = -w-w = These are also known as COLUMN VECTORS C D v E F w

C D v E F w v == This is also known as the components of v

Q P a b It can be calculated using Pythagoras ExampleIf

Vectors are only equal if they have the same magnitude and direction. Equal Vectors a b c d For vectors and

For vectors and Addition Of Vectors

== i) Find the components of = =

The Zero Vector is called the zero vector written 0 If find the components of

Subtraction of Vectors Page 236 Exercise 13D questions 1 to 3

Multiplication by a scalar

Unit Vectors For any vector v there exists a parallel vector u of magnitude 1 unit. This is called a Unit Vector. i.e. Find the components of the unit vector u parallel to vector Since the magnitude of v is 5, the unit vector u must be 1 / 5 v

Position Vectors If P and Q have coordinates (4,8) and (2,3) find the components of

Collinearity We have seen that if a vector v = ku then v must be parallel to u. If vectors v and u also have a point in common then because they are parallel they must lie on the same line so by definition must be collinear.

Prove that the points A(2,4), B(8,6) and C(11,7) are collinear. B is a point in common to both AB and BC so A, B and C are collinear

Section Formula If p is the position vector of the point P that divides AB in the ratio m:n then: A B P m n

A and B have coordinates (3,2) and (7,14) respectively. Find the coordinates of the point P that divides AB in the ratio 1:3 1. Draw a quick sketch (3,2) (7,14) P 1 3

3 Dimensional Vectors x y z A The point A has a position relative to the x y and z axis A(3, 4, 6)

Find the coordinates of P y x z P P(4, 2, 1) O

Find the coordinates of Q y x z Q Q(-1, -2, -3)

3D Unit Vectors A vector can also be defined in terms of i, j and k where i, j and k are unit vectors in the x, y, and z directions respectively. y x z i 1 j 1 1k1k In component form the vectors are written as

Any vector can be expressed as a combination of its components.

Properties of 3D vectors P Q R S

Addition / Subtraction Scalar Position Vector

Section Formula A (4, -6, 12) B (4, 4, -3) P 3 2

The Scalar Product For two vectors a and b, the scalar product is defined by Where  is the angle between a and b, The scalar product is also known as the dot product. The vectors must be directed away from the point of intersection.

If a and b are perpendicular then a. b = 0

Component Form of a Scalar Product

Angle between vectors

Other Vector Facts PROOF

30 0 p r q