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Addition of vectors (i) Triangle Rule [For vectors with a common point] C B A

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**(ii) Parallelogram Rule [for vectors with same initial point]**

D C B A

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**(iii) Extensions follow to three or more vectors**

p+q+r q p

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**First we need to understand what is meant by the vector – a**

Subtraction First we need to understand what is meant by the vector – a a – a a and – a are vectors of the same magnitude, are parallel, but act in opposite senses.

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A few examples b – a a a b b

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**q p p q – q p Which vector is represented by p + q ?**

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B CB = CA + AB = - AC + AB = AB – AC C A

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**OP is a position vector of a point P. We usually associate p with OP**

Position Vectors Relative to a fixed point O [origin] the position of a Point P in space is uniquely determined by OP P p OP is a position vector of a point P. We usually associate p with OP O

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**A very Important result!**

B AB = b - a b A a O

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The Midpoint of AB A M OM = ½(b + a) a B b O

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**Vectors Questions P A a N O b B Q Example**

In the diagram, OA = AP and BQ = 3OB. N is the midpoint of PQ. and A P N Q O B a b Express each of the following vectors in terms of a, b or a and b. (a) (b) (c) (d) (e) (f) (g) (h)

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OA = AP and BQ = 3OB a) A P N Q O B a b b) c) d) e) f) g) h)

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Example ABCDEF is a regular hexagon with , representing the vector m and , representing the vector n. Find the vector representing m A B C D F E n m+n m

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**Example M, N, P and Q are the mid-points of OA, OB, AC and BC.**

OA = a, OB = b, OC = c (a) Find, in terms of a, b and c expressions for (i) BC (ii) NQ (iii) MP (b) What can you deduce about the quadrilateral MNQP? a) BC = BO + OC = c – b (ii) NQ = NB + BQ b = c a (ii) MP = MA + AP c = c MNPQ is a parallelogram as NQ and MP are equal and parallel.

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Example In the diagram, and , OC = CA, OB = BE and BD : DA has ratio 1 : 2. a) Express in terms of a and b (i) b) Explain why points C, D and E lie on a straight line. O D B C E A b a

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Example ABC is a triangle with D the midpoint of BC and E a point on AC such that AE : EC = 2 : 1. Prove that the sum of the vectors , , is parallel to

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