Shape and Space 9. Vectors Mr F’s Maths Notes Shape and Space 9. Vectors

9. Vectors b a a b c d c d 1. What are Vectors? Vectors are just a posh (and quite convenient) way of describing how to get from one point to another Starting from the tail of the vector, the number on the top tells you how far right/left to go, and the number on the bottom tells your how far up/down If this number is positive, you move right, if it is negative, you move left If this number is positive, you move up, if it is negative, you move down b a 1 to the right, and 3 up a 5 to the right, and 2 down b c d c 3 to the left, and 2 down d 0 to the right, and 3 up

5 a a 2 b b 4 3 2. The Magnitude of Vectors By forming right-angled triangles and using Pythagoras’ Theorem, it is possible to work out the magnitude (size) of any vector 5 a a 2 b b 4 3 Note: Because you are squaring the numbers, you do not need to worrying about negatives!

b a + b a b a a + b c + d d c d c c + d 3. Adding Vectors When you add two or more vectors together, you simply add the tops and add the bottoms The new vector you end up with is called the resultant vector b a + b a b a a + b c + d d c d c c + d Watch Out! Remember to be careful with your negatives!

- a a - a a q - q p p q p p - q p – q = p + (-q) 4. Subtracting Vectors The negative of a vector goes in the exact opposite direction, which changes the signs of the numbers on the top and the bottom (see below) One way to think about subtracting vectors is to simply add the negative of the vector! - a a - a a q - q p p q p p - q p – q = p + (-q)

p p 2p 2p q 3q q 3q r - 4r r -4r 5. Multiplying Vectors The only thing you need to remember when multiplying vectors is that you multiply both the top and the bottom of the vector! p p 2p 2p q 3q q 3q r - 4r r -4r

(a) 4a + 3b + c (a) 2a - 5b - 2c 6. Linear Combinations of Vectors Using the skills we learnt when multiplying vectors, it is possible to calculate some pretty complicated looking combinations of vectors Example: If Calculate the following: (a) 4a + 3b + c (a) 2a - 5b - 2c Watch Out! Remember to be so, so careful with your negatives!

a  B A (i) FC b  FC = 2a O F C  (ii) DA E D  DA = – 2b 7. Vectors in Geometry A popular question asked by the lovely examiners is to give you a shape and ask you to describe a route between two points using vectors. There is one absolutely crucial rule here… you can only travel along a route of known vectors! Just because a line looks like it should be a certain vector, doesn’t mean it is! Example: Below is a regular hexagon. Describe the routes given in terms of vectors a and b a (i) FC  A B The best way to go here is straight across the middle, because we know each horizontal line is just a b  FC = 2a O F C (ii) DA  E Again, the middle is looking good here, but remember we are going the opposite way to our given vector, so we need the negative! D  DA = – 2b

 a B (iii) EB A b O F      C EB = EF + FO + OA + AB It would be nice to just nip across the middle, but the problem is we do not know what those vectors are! So… we’ll just have to go the long way around, travelling along routes we do know! b O F      C EB = EF + FO + OA + AB = -b + a + -b + a = 2a – 2b E D

Good luck with your revision!