Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com GCSE: Vectors Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Last modified: 20th March 2017

Coordinates vs Vectors 𝑦 6 ? Coordinates represent a position. We write 𝑥,𝑦 to indicate the 𝑥 position and 𝑦 position. 5 4 ? ? 2,3 Vectors represent a movement. We write 𝑥 𝑦 to indicate the change in 𝑥 and change in 𝑦. 3 ? 4 1 2 Note we put the numbers vertically. Do NOT write 4 1 ; there is no line and vectors are very different to fractions! You may have seen vectors briefly before if you’ve done transformations; we can use them to describe the movement of a shape in a translation. 1 𝑥 1 2 3 4 5 6 7 -1 -2

Vectors to represent movements 𝒃 𝒂 𝒂= 1 3 𝒃= 5 1 𝒄= 2 −3 𝒅= −4 2 𝒆= −4 −4 𝒇= −1 1 𝒈= 0 −3 ? ? 𝒅 ? ? 𝒇 ? ? 𝒈 𝒆 ? 𝒄

Writing Vectors You’re used to variables representing numbers in maths. They can also represent vectors! 𝑌 What can you say about how we use variables for vertices (points) vs variables for vectors? We use capital letters for vertices and lower case letters for vectors. There’s 3 ways in which can represent the vector from point 𝑋 to 𝑍: 𝒂 (in bold) 𝑎 (with an ‘underbar’) 𝑋𝑍 𝒃 𝑍 ? 𝒂 ? ? 𝑋 ?

Adding Vectors ? ? 𝑌 ? 𝑍 ? 𝑋 𝒃 𝒂 𝑋𝑍 =𝒂= 2 5 𝑍𝑌 =𝒃= 5 3 𝑋𝑌 = 7 8 𝑋𝑍 =𝒂= 2 5 𝑍𝑌 =𝒃= 5 3 𝑋𝑌 = 7 8 ? ? 𝑌 ? 𝒃 What do you notice about the numbers in 7 8 when compared to 2 5 and 5 3 ? We’ve simply added the 𝑥 values and 𝑦 values to describe the combined movement. i.e. 𝒂+𝒃= 2 5 + 5 3 = 7 8 𝑋𝑍 + 𝑍𝑌 = 𝑋𝑌 𝑍 ? 𝒂 Bro Important Note: The point is that we can use any route to get from the start to finish, and the vector will always be the same. Route 1: We go from 𝑋 to 𝑌 via 𝑍. 𝑋𝑍 = 2 5 + 5 3 = 7 8 Route 2: Use the direct line from 𝑋 to 𝑌: 7 8 𝑋

Scaling Vectors We can ‘scale’ a vector by multiplying it by a normal number, aptly known as a scalar. If 𝒂= 4 3 , then 2𝒂=2 4 3 = 8 6 What is the same about 𝒂 and 2𝒂 and what is different? ? 𝒂 2𝒂 Same: Same direction / Parallel Different: The length of the vector, known as the magnitude, is longer. ? ? Bro Note: Note that vector letters are bold but scalars are not.

More on Adding/Subtracting Vectors 𝑋 If 𝑂𝐴 =𝒂, 𝐴𝐵 =𝒃 and 𝑋𝐵 =2𝒄, then find the following in terms of 𝑎, 𝑏 and 𝑐: 𝑂𝐵 =𝒂+𝒃 𝑂𝑌 =𝒂+2𝒃 𝐴𝑋 =𝒃−2𝒄 𝑋𝑂 =2𝒄−𝒃−𝒂 𝑌𝑋 =−𝒃−2𝒄 2𝒄 𝐴 𝐵 𝒃 𝒃 𝑌 ? 𝒂 ? 𝒂 ? 𝑂 ? 𝒃 ? Note: Since − 𝑥 𝑦 = −𝑥 −𝑦 , subtracting a vector goes in the opposite direction. Bro Side Note: Since 𝑏+𝑎 would end up at the same finish point, we can see 𝒃+𝒂=𝒂+𝒃 (i.e. vector addition, like normal addition, is ‘commutative’)

Exercise 1 (on provided sheet) 1 State the value of each vector. 2 4 𝒂 𝒃 𝐵𝐴 =−𝑎 𝐴𝐶 =𝑎+𝑏 𝐷𝐵 =2𝑎−𝑏 𝐴𝐷 =−𝑎+𝑏 ? 𝑍𝑋 =𝑎+𝑏 𝑌𝑊 =𝑎−2𝑏 𝑋𝑌 = −𝑎+𝑏 𝑋𝑍 =−𝑎−𝑏 ? 𝒄 ? ? ? ? ? ? 𝒅 3 5 𝒆 𝒇 𝑀𝐾 =−𝑎−𝑏 𝑁𝐿 =3𝑎−𝑏 𝑁𝐾 =2𝑎−𝑏 𝐾𝑁 =−2𝑎+𝑏 ? ? ? 𝐴𝐵 =−𝑎+𝑏 𝐹𝑂 =−𝑎+𝑏 𝐴𝑂 =−2𝑎+𝑏 𝐹𝐷 =−3𝑎+2𝑏 ? ? 𝒂= 3 1 𝒃= 3 −2 𝒄= −3 0 𝒅= 0 5 𝒆= −1 4 𝒇= −3 −4 ? ? ? ? ? ? ? ? ?

Starter Expand and simplify the following expressions: ? 2 𝒂+𝒃 +𝒃 =2𝒂+3𝒃 1 2 𝒂−𝒃 +𝒃 = 1 2 𝒂+ 1 2 𝒃 𝒂+ 1 2 𝒂+𝒃 = 3 2 𝒂+ 1 2 𝒃 𝒂−2 𝒂−2𝒃 =−𝒂+4𝒃 1 3 𝒂+2𝒃 −𝒃= 1 3 𝒂− 1 3 𝒃 1 ? 2 ? 3 ? 4 ? 5

The ‘Two Parter’ exam question Many exams questions follow a two-part format: Find a relatively easy vector using skills from Lesson 1. Find a harder vector that uses a fraction of your vector from part (a). Edexcel June 2013 1H Q27 Bro Tip: This ratio wasn’t in the original diagram. I like to add the ratio as a visual aid. 𝑆𝑄 =−𝒃+𝒂 ? a For (b), there’s two possible paths to get from 𝑁 to 𝑅: via 𝑆 or via 𝑄. But which is best? In (a) we found 𝑺 to 𝑸 rather than 𝑸 to 𝑺, so it makes sense to go in this direction so that we can use our result in (a). 3 : 2 ? ? 𝑁𝑅 = 2 5 𝑆𝑄 +𝒃 = 2 5 −𝒃+𝒂 +𝒃 = 2 5 𝒂+ 3 5 𝒃 𝑄𝑅 is also 𝑏 because it is exactly the same movement as 𝑃𝑆 . b Bro Workings Tip: While you’re welcome to start your working with the second line, I recommend the first line so that your chosen route is clearer.

Test Your Understanding Edexcel June 2012 𝐴𝐵 = 𝐴𝑂 + 𝑂𝐵 =−𝒂+𝒃 ? 𝑂𝑃 =𝒂+ 3 4 𝐴𝐵 =𝒂+ 3 4 −𝒂+𝒃 = 1 4 𝒂+ 3 4 𝒃 ? You MUST expand and simplify.

Further Test Your Understanding 𝐵 𝐵 B 𝑂𝑋:𝑋𝐵=1:3 𝐴𝑌:𝑌𝐵=2:3 𝒃 𝑋 𝒃 𝑌 𝑂 𝒂 𝐴 𝑂 𝒂 𝐴 𝐴𝑋 =−𝒂+ 1 4 𝑂𝐵 =−𝒂+ 1 4 𝒂+𝒃 =−𝒂+ 1 4 𝒂+ 1 4 𝒃 =− 3 4 𝒂+ 1 4 𝒃 First Step? 𝑂𝑌 =𝒂+ 2 5 𝐴𝐵 =𝒂+ 2 5 −𝒂+𝒃 =𝒂− 2 5 𝒂+ 2 5 𝒃 = 3 5 𝒂+ 2 5 𝒃 First Step? ? ?

Exercise 2 ? ? ? ? ? ? ? ? ? (on provided sheet) 3 1 𝑋 is a point such that 𝐴𝑋:𝑋𝐵=1:4 3 𝐴𝐵 =−𝒂+𝒃 𝐴𝑋 =− 1 5 𝒂+ 1 5 𝒃 𝑂𝑋 = 4 5 𝒂+ 1 5 𝒃 𝐵𝑋 = 4 5 𝒂− 4 5 𝒃 ? ? ? [June 2009 2H Q23] a) Find 𝐴𝐵 in terms of 𝒂 and 𝒃. 𝐴𝐵 =−𝒂+𝒃 𝑜𝑟 𝒃−𝒂 b) 𝑃 is on 𝐴𝐵 such that 𝐴𝑃:𝑃𝐵=3:2. Show that 𝑂𝑃 = 1 5 2𝒂+3𝒃 ? ? 2 𝑌 is a point such that 𝑌𝐵=2𝐴𝑌 ? 𝐴𝑌 =− 1 3 𝒂+ 1 3 𝒃 𝑂𝑌 = 2 3 𝒂+ 1 3 𝒃 𝑌𝑂 =− 2 3 𝒂− 1 3 𝒃 ? ? ?

Exercise 2 ? ? ? ? ? ? ? ? ? ? ? ? (on provided sheet) [Nov 2010 1H Q27] 𝑀 is the midpoint of 𝑂𝑃. 𝑂𝐴𝐶𝐵 is a parallelogram. 𝑅 is a point such that 𝐴𝑅:𝑅𝐵=2:3. 𝑆 is a point such that 𝐵𝑆:𝑆𝐶=1:3. 4 5 ? 𝑂𝑅 = 3 5 𝒂+ 2 5 𝒃 𝐵𝑆 = 1 4 𝒂 𝑂𝑆 = 1 4 𝒂+𝒃 𝑅𝑆 =− 7 20 𝒂+ 3 5 𝒃 ? ? ? 6 Express 𝑂𝑀 in terms of 𝒂 and 𝒃. 𝑂𝑀 = 1 2 𝒂+𝒃 𝑜𝑟 1 2 𝒂+ 1 2 𝒃 Express 𝑇𝑀 in terms of 𝒂 and 𝒃 giving your answer in its simplest form. 𝑇𝑀 =−𝒂+ 1 2 𝑂𝑀 =−𝒂+ 1 2 𝒂+𝒃 = 1 2 𝒃− 1 2 𝒂 𝑀 is the midpoint of 𝐶𝐷, 𝐵𝑃:𝑃𝑀=2:1 ? 𝐷𝐶 =−𝒛+𝒚 𝐷𝑀 =− 1 2 𝒛+ 1 2 𝒚 𝐴𝑀 = 𝐴𝐷 + 𝐷𝑀 = 1 2 𝒛+ 1 2 𝒚 𝐵𝑀 = 𝐵𝐴 + 𝐴𝑀 =−𝒙+ 1 2 𝒛+ 1 2 𝒚 𝐵𝑃 = 2 3 𝐵𝑀 =− 2 3 𝒙+ 1 3 𝒛+ 1 3 𝒚 𝐴𝑃 = 𝐴𝐵 + 𝐵𝑃 = 1 3 𝒙+ 1 3 𝒛+ 1 3 𝒚 ? ? ? ? ? ? ?

Exercise 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? (on provided sheet) 8 7 𝐴𝐵 =−𝒂+𝒃 𝐵𝐶 =𝒂 𝑀𝐵 =− 1 2 𝒂+ 1 2 𝒃 𝑀𝑋 =− 1 6 𝒂+ 1 2 𝒃 𝑋𝐴 = 2 3 𝒂−𝒃 𝐶𝑀 =− 1 2 𝒂− 1 2 𝒃 𝑋𝑂 =− 1 3 𝒂−𝒃 ? 𝑂𝐵 =𝒂+2𝒃 𝐵𝐶 =𝒂−𝒃 𝐴𝑀 = 1 2 𝒂+𝒃 𝑂𝑀 = 3 2 𝒂+𝒃 ? ? ? ? ? ? ? ? ? 9 ? 𝑂𝐶 =−𝒂+𝒃 𝑋𝐶 =− 3 4 𝒂+ 1 2 𝒃 𝐴𝑋 =− 5 4 𝒂+ 1 2 𝒃 ? ? ?

Recap ?

Types of vector ‘proof’ questions “Show that … is parallel to …” “Prove these two vectors are equal.” “Prove that … is a straight line.”

Showing vectors are equal Edexcel March 2013 ? ? 2 5 𝑂𝑌 = 2 5 6𝑏+5𝑎−𝑏 =2𝑎+2𝑏 𝑂𝑋 =3𝑎+ 1 3 −3𝑎+6𝑏 =2𝑎+2𝑏 ∴ 𝑂𝑋 = 2 5 𝑂𝑌 There is nothing mysterious about this kind of ‘prove’ question. Just find any vectors involved, in this case, 𝑂𝑋 and 2 5 𝑂𝑌 . Hopefully they should be equal! With proof questions you should restate the thing you are trying to prove, as a ‘conclusion’.

What do you notice? ? 𝒂+2𝒃 𝒂 2𝒂 2𝒂+4𝒃 ! Vectors are parallel if one is a multiple of another. 𝒂 − 3 2 𝒂

Parallel or not parallel? Vector 1 Vector 2 Parallel? (in general) 𝑎+𝑏 𝑎+2𝑏 3𝑎+3𝑏 −3𝑎−6𝑏 𝑎−𝑏 −𝑎+𝑏 No  Yes   No Yes   No Yes  No   Yes For ones which are parallel, show it diagrammatically.

How to show two vectors are parallel 𝑨 𝑪 𝒂 𝑿 𝑴 𝑶 𝑩 𝒃 𝑋 is a point on 𝐴𝐵 such that 𝐴𝑋:𝑋𝐵=3:1. 𝑀 is the midpoint of 𝐵𝐶. Show that 𝑋𝑀 is parallel to 𝑂𝐶 . ? For any proof question always find the vectors involved first, in this case 𝑋𝑀 and 𝑂𝐶 . 𝑂𝐶 =𝑎+𝑏 𝑋𝑀 = 1 4 −𝑎+𝑏 + 1 2 𝑎= 1 4 𝑎+ 1 4 𝑏 = 1 4 𝑎+𝑏 𝑋𝑀 is a multiple of 𝑂𝐶 ∴ parallel. The key is to factor out a scalar such that we see the same vector. The magic words here are “is a multiple of”.

Test Your Understanding Edexcel June 2011 Q26 𝐴 Find 𝐴𝐵 in terms of 𝒂 and 𝒃. −2𝒂+3𝒃 𝑃 is the point on 𝐴𝐵 such that 𝐴𝑃:𝑃𝐵=2:3. Show that 𝑂𝑃 is parallel to the vector 𝒂+𝒃. 𝑃 ? 2𝒂 𝐵 𝑂 3𝒃 ? Bro Exam Note: Notice that the mark scheme didn’t specifically require “is a multiple of” here (but write it anyway!), but DID explicitly factorise out the 6 5

Proving three points form a straight line Points A, B and C form a straight line if: 𝑨𝑩 and 𝑩𝑪 are parallel (and B is a common point). Alternatively, we could show 𝑨𝑩 and 𝑨𝑪 are parallel. This tends to be easier. ? C B A C B A

Straight Line Example ? ? a −3𝒃+𝒂 b 𝐴𝑁 =2𝒃, 𝑁𝑃 =𝒃 𝑁𝑀 =𝑏+ 1 2 𝑎−3𝑏 =𝑏+ 1 2 𝑎− 3 2 𝑏 = 1 2 𝑎− 1 2 𝑏 = 𝟏 𝟐 (𝒂−𝒃) 𝑁𝐶 =−2𝑏+2𝑎 =𝟐 𝒂−𝒃 𝑵𝑪 is a multiple of 𝑵𝑴 ∴ 𝑵𝑪 is parallel to 𝑵𝑴 . 𝑵 is a common point. ∴ 𝑵𝑴𝑪 is a straight line. b 𝐴𝑁 =2𝒃, 𝑁𝑃 =𝒃 𝐵 is the midpoint of 𝐴𝐶. 𝑀 is the midpoint of 𝑃𝐵. a) Find 𝑃𝐵 in terms of 𝒂 and 𝒃. b) Show that 𝑁𝑀𝐶 is a straight line.

Test Your Understanding November 2013 1H Q24 𝑂𝐴 =𝑎 and 𝑂𝐵 =𝑏 𝐷 is the point such that 𝐴𝐶 = 𝐶𝐷 The point 𝑁 divides 𝐴𝐵 in the ratio 2:1. (a) Write an expression for 𝑂𝑁 in terms of 𝒂 and 𝒃. 𝑶𝑵 =𝒃+ 𝟏 𝟑 −𝒃+𝒂 = 𝟏 𝟑 𝒂+ 𝟐 𝟑 𝒃 (b) Prove that 𝑂𝑁𝐷 is a straight line. 𝑶𝑫 =𝒂+𝟐𝒃 𝑶𝑵 = 𝟏 𝟑 (𝒂+𝟐𝒃) 𝑶𝑵 is a multiple of 𝑶𝑫 and 𝑶 is a common point. ∴ 𝑶𝑵𝑫 is a straight line. ? ?

Exercise 3 1 2 𝑇 is the point on 𝐴𝐵 such that 𝐴𝑇:𝑇𝐵=5:1. Show that 𝑂𝑇 is parallel to the vector 𝒂+2𝒃. 𝑂𝑇 =2𝒃+ 1 6 −2𝒃+5𝒂 =2𝒃− 1 3 𝒃+ 5 6 𝒂 = 5 6 𝒂+ 5 3 𝒃 = 5 6 𝒂+2𝒃 𝑀 is the midpoint of 𝑂𝐴. 𝑁 is the midpoint of 𝑂𝐵. Prove that 𝐴𝐵 is parallel to 𝑀𝑁. 𝐴𝐵 =−2𝒎+2𝒏 =2 −𝒎+𝒏 𝑀𝑁 =−𝒎+𝒏 𝐴𝐵 is a multiple of 𝑀𝑁 ∴ parallel. ? ?

Exercise 3 3 4 𝐴𝐶𝐸𝐹 is a parallelogram. 𝐵 is the midpoint of 𝐴𝐶. 𝑀 is the midpoint of 𝐵𝐸. Show that 𝐴𝑀𝐷 is a straight line. 𝐷𝑀 =−𝒃+ 1 2 3𝒃+𝒂 =−𝒃+ 3 2 𝒃+ 1 2 𝒂 = 1 2 𝒂+ 1 2 𝒃= 1 2 𝒂+𝒃 𝐷𝐴 =2𝒃+2𝒂=2 𝒂+𝒃 𝐷𝐴 is a multiple of 𝐷𝑀 and 𝐷 is a common point, so 𝐴𝑀𝐷 is a straight line. 𝐶𝐷 =𝒂, 𝐷𝐸 =𝒃 and 𝐹𝐶 =𝒂−𝒃 i) Express 𝐶𝐸 in terms of 𝒂 and 𝒃. 𝒂+𝒃 ii) Prove that 𝐹𝐸 is parallel to 𝐶𝐷 . 𝐹𝐸 =𝑎−𝑏+𝑎+𝑏=2𝒂 which is a multiple of 𝐶𝐷 iii) 𝑋 is the point on 𝐹𝑀 such that such that 𝐹𝑋:𝑋𝑀=4:1. Prove that 𝐶, 𝑋 and 𝐸 lie on the same straight line. 𝐶𝑋 =−𝒂+𝒃+ 4 5 2𝒂− 1 2 𝒃 = 3 5 𝒂+ 3 5 𝒃 = 3 5 𝒂+𝒃 𝐶𝐸 =𝒂+𝒃 ? ? ? ?

Exercise 3 5 6 𝑂𝐴 =3𝒂 and 𝐴𝑄 =𝒂 and 𝑂𝐵 =𝒃 and 𝐵𝐶 = 1 2 𝒃. 𝑀is the midpoint of 𝑄𝐵. Prove that 𝐴𝑀𝐶 is a straight line. 𝐴𝑀 =𝒂+ 1 2 −4𝒂+𝒃 =−𝒂+ 1 2 𝒃 𝐴𝐶 =−3𝒂+ 3 2 𝒃=3 −𝒂+ 1 2 𝒃 𝐴𝐶 is a multiple of 𝐴𝑀 and 𝐴 is a common point, therefore 𝐴𝑀𝐶 is a straight line. 𝑂𝐴𝐵𝐶 is a parallelogram. 𝑃 is the point on 𝐴𝐶 such that 𝐴𝑃= 2 3 𝐴𝐶. i) Find the vector 𝑂𝑃 . Give your answer in terms of 𝒂 and 𝒄. 𝑂𝑃 =6𝒂+ 2 3 −6𝒂+6𝒄 =2𝒂+4𝒄=2 𝒂+2𝒄 ii) Given that the midpoint of 𝐶𝐵 is 𝑀, prove that 𝑂𝑃𝑀 is a straight line. 𝑂𝑀 =3𝒂+6𝒄=3 𝒂+2𝒄 𝑂𝑀 is a multiple of 𝑂𝑃 and 𝑂 is a common point, therefore 𝑂𝑃𝑀 is a straight line. ? ? ?