CIE Centre A-level Pure Maths P1 Chapter 13 CIE Centre A-level Pure Maths © Adam Gibson

scalars matrices vectors operators geometrical objects tensors

VECTORS A vector is a mathematical object which has two properties: SIZE & direction Examples of vector quantities: velocity, force, distance, temperature, torque

Vectors in 2D – the basics We can write a vector in the x-y plane as a certain number of x-units and a certain number of y-units. When you write vectors, you should underline them 4 3 This vector is specified by 3 units in the x-direction and 4 units in the y-direction. We can write it as or as ; 3 and 4 are the components In the x- and y-directions.

Vectors in 2D – the basics - magnitude The magnitude of a vector is another name for its size. What is the size of this vector? 4 3 It can be calculated using Pythagoras’ theorem. The size is the same as the length, which is

Vectors in 2D – the basics - location ? Question: Where is the vector Answer: everywhere! A vector can represent a simple translation

Unit vectors A unit vector can be calculated in any direction by dividing by the magnitude. 1 unit 1 unit Every vector in the x-y plane can be expressed as a linear combination of i and j. Write the vector from (2,7) to (8,-5) in terms of i and j. Calculate the unit vector in the direction of the vector

Vector arithmetic The rules for adding and subtracting vectors are easy.. The Parallelogram Rule… Now look at Fig 13.5 p. 191 means that

Vector arithmetic – continued. Hence you should see that: “the commutative rule for the addition of vectors is a consequence of the commutative rule for the addition of real numbers”. If , what is the magnitude of p + q ? Subtraction is no different: Calculate the magnitude of and illustrate your calculation on a graph.

Vector arithmetic – continued. Q: We can add and subtract vectors, but can we multiply or divide them? The answer is yes and no. Multiplication by a scalar. Illustrate, on a graph, the meaning of How is the picture different if s < 0 or s > 0? Now try Q 3, 5 and 10 on page 194.

Vectors – 3D Vectors in 3 dimensions are very useful for solving real world (or virtual world!) problems.

3D vs 2D compared. Q: Consider everything we’ve learnt about lines, equations and calculus. Is it different in a 3 dimensional coordinate system? Non-parallel lines don’t always cross. Such lines are called skew. Gradient – it’s not defined purely by a number. The direction of a line is only defined by a vector. Planes in 3D are the equivalent of lines in 2D. Non-parallel planes always do cross. We will study how to represent planes in vector form later.

The dot product gives a quantitative measure The “scalar product” or “dot product” The scalar product is defined, for any 2 vectors p and q, as: Why do we call it a “scalar” product? What is |p|? What is the angle θ? If x=24 and p=3i+2j, what is x.p? What is the real meaning of the dot product? The dot product gives a quantitative measure of the extent to which 2 vectors are collinear.

The “scalar product” or “dot product” - continued The dot product a.b is.. zero b a b The dot product a.b is.. |a||b| a If |a|=|b|=1, then the dot product a.b is.. b 0.5 a

The “scalar product” or “dot product” - continued What does it mean if p.q = 0? The vectors are perpendicular. What is p.p? It’s the square of the magnitude Of the vector, |p|2 In component form Assume the distributive rule: See p. 205 – you don’t need to prove this Then remembering i.i = j.j = k.k = 1 and i.j = j.k = i.k = 0,

The “scalar product” or “dot product” - continued We thereby get the very useful result: Which means we don’t need to use trigonometric functions to calculate dot products. In particular we can see that a.a = |a|2 implies: (note that this can be extended to any number of dimensions!)

TASKS Find the magnitude of each of the following vectors. Which vectors are perpendicular? What is the angle between p and q (in radians, 1d.p.)? **Can you find a vector skew to all these four? A: |p| = √14, |q|=√369/2,|r|= √19,s=2 √5 p and r are perpendicular, p and s are perpendicular, Angle between p and q is: cos-10.5426 = 1.0 radian

Relative Velocity – Key Principles Know how to resolve vectors Know and understand the key equation for relative velocity: “The velocity of b relative to a is the velocity of b minus the velocity of a”

Problems with Relative Velocity N W E S New York 5600km London A plane flies directly West from London to New York, a distance of 5600km. Its maximum speed through still air is 950 km/hr. The wind speed is 120 km/hr at a bearing of 20 degrees. How quickly can it reach New York, and at what bearing does the plane fly relative to the air?

Problems with Relative Velocity N W E S New York 5600km London

Problems with Relative Velocity The velocity of the plane relative to the air is the true velocity of the plane minus the velocity of the air land water speed It’s the same for a boat. The boat’s direction in the water must be towards the left (opposing the water) if we want to travel directly to the other side.

Problems with Relative Velocity We need to resolve the wind (air) speed into two components. 20 The aircraft must travel due West. So what is ?

Problems with Relative Velocity We know that the magnitude of is 950. So we can draw like this:

Problems with Relative Velocity To make the problem simpler, we define unit vectors i and j in the directions West and North respectively.

Problems with Relative Velocity At what angle is the plane flying relative to the air? N Bearing =

Problems with Relative Velocity A traveller on a train travelling with velocity sees a car out of the window, which appears to be travelling at velocity What is the true speed of the car? True velocity of car is given by: