T Madas A vector is a line with

A vector is a line with a start and a finish. It therefore has:

we write vectors in the following ways: By writing the starting point and the

ABCD is a parallelogram B b a A C a b D © T

ABCD is a parallelogram B b a A C a b D Now adding

ABC is a triangle with M the midpoint of AB and N the midpoint

ABCDEF is a regular hexagon. M is the midpoint of CE. Write and simplify

ABCD is a parallelogram. M is the midpoint of AD N is a point

ABCD is a parallelogram. M is the midpoint of AB N is a point

ABCD is a quadrilateral and M, N , P and Q are the midpoints

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A vector is a line with a start and a finish. It therefore has: 1. 2. 3. line of action a direction a given size (magnitude) A B © T Madas

we write vectors in the following ways: By writing the starting point and the finishing point in capitals with an arrow over them With a lower case letter which: is printed in bold or underlined when handwritten In component form if the vector is drawn on a grid: 4 5 © T Madas

F E B D H A C G © T Madas

B D AB = 4 5 A -5 CD = 4 C © T Madas

ABCD is a parallelogram B b a A C a b D © T Madas

ABCD is a parallelogram B b a A C a b D Now adding vectors © T Madas

ABC is a triangle with M the midpoint of AB and N the midpoint of BC. B M A N C © T Madas

ABC is a triangle with M the midpoint of AB and N the midpoint of BC. B M A N C What is the relationship between AC and MN ? © T Madas

ABC is a triangle with M the midpoint of AB and N the midpoint of BC. B M A N C © T Madas

ABCDEF is a regular hexagon. M is the midpoint of CE. Write and simplify expressions in terms of a, b and c for : D C M B E A F solution © T Madas

ABCD is a parallelogram. M is the midpoint of AD N is a point of BD so that BN : ND = 2 : 1. Write and simplify expressions in terms of a and b for : B C solution N A M D © T Madas

ABCD is a parallelogram. M is the midpoint of AD N is a point of BD so that BN : ND = 2 : 1. Write and simplify expressions in terms of a and b for : (e) Using your answers from parts (c) and (d), show that M, N and C lie on a straight line B C solution N A M D © T Madas

ABCD is a parallelogram. M is the midpoint of AB N is a point of BD so that. (a) (b) Find the vector in terms of a and b Prove that MNC is a straight line C B N M A D © T Madas

ABCD is a quadrilateral and M, N , P and Q are the midpoints of AB, BC, CD and DA respectively. B AM = a, BN = b and CP = c. b N Find in terms of a, b and c: b a C a) AD b) AQ M c) MQ a P d) NP c e) Deduce a geometric fact A a+b+c Q a+b+c D about the quadrilateral MNPQ c AD = AB + BC + CD = 2 a + 2 b + 2 c = 2(a + b + c) AQ = a + b + c © T Madas

ABCD is a quadrilateral and M, N , P and Q are the midpoints of AB, BC, CD and DA respectively. B AM = a, BN = b and CP = c. b N Find in terms of a, b and c: b a C a) AD b) AQ M b+c c) MQ b+c a P d) NP c e) Deduce a geometric fact A a+b+c Q a+b+c D about the quadrilateral MNPQ c MQ = MA + AQ = -a + (a + b + c) = b + c NP = NC + CP = b + c © T Madas

ABCD is a quadrilateral and M, N , P and Q are the midpoints of AB, BC, CD and DA respectively. B AM = a, BN = b and CP = c. b N Find in terms of a, b and c: b a C a) AD b) AQ M b+c c) MQ b+c a P d) NP c e) Deduce a geometric fact A a+b+c Q a+b+c D about the quadrilateral MNPQ c A quadrilateral with a pair of sides equal and parallel is a parallelogram or can show that MN = QP = a + b Hence MNPQ is a parallelogram. © T Madas