1. Introducing Vectors
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2. A vector is a line with a start and a finish. It therefore has:
line of action
a direction
a given size (magnitude) B A B A
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3. 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
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4. D F B E H A C G
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5. B D 4 5
AB = -5 4
CD = A C
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7. ABCD is a parallelogram
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8. ABCD is a parallelogram
Now adding vectors
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9. ABCD is a parallelogram
Now adding vectors
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11. with M the midpoint of AB and N the midpoint of BC.
ABC is a triangle
with M the midpoint of AB
and N the
midpoint of BC. B M N A C
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12. with M the midpoint of AB and N the midpoint of BC.
ABC is a triangle
with M the midpoint of AB
and N the
midpoint of BC. B M N A C
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13. What is the relationship between AC and MN ?
ABC is a triangle
with M the midpoint of AB
and N the
midpoint of BC. B M N A C
What is the relationship between AC and MN ?
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14. with M the midpoint of AB and N the midpoint of BC.
ABC is a triangle
with M the midpoint of AB
and N the
midpoint of BC. B M N A C
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16. ABCDEF is a regular hexagon. M is the midpoint of CE.
Write and simplify expressions in terms of a, b and c for : C D M B E A F
solution
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17. ABCDEF is a regular hexagon. M is the midpoint of CE.
Write and simplify expressions in terms of a, b and c for : C D M B E A F
solution
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19. 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 D M
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20. 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 D M
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21. 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 D M
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22. What is the ratio MN : NC ? 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 D M
What is the
ratio MN : NC ?
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24. ABCD is a parallelogram. M is the midpoint of AB
N is a point of BD so that
(a) Find the vector in terms of a and b
(b) Prove that MNC is a straight line B C N M D A
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25. ABCD is a parallelogram. M is the midpoint of AB
N is a point of BD so that
(a) Find the vector in terms of a and b
(b) Prove that MNC is a straight line B C N M D A
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26. ABCD is a parallelogram. M is the midpoint of AB
N is a point of BD so that
(a) Find the vector in terms of a and b
(b) Prove that MNC is a straight line B C N M D A
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27. ABCD is a parallelogram. M is the midpoint of AB
N is a point of BD so that
(a) Find the vector in terms of a and b
(b) Prove that MNC is a straight line B C N M D A
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29. Find in terms of a, b and c: AD AQ MQ NP
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.
Find in terms of a, b and c: AD AQ MQ NP
Deduce a geometric fact about the quadrilateral MNPQ b N b a C M c a P c A Q
a + b + c
a + b + c D
AD = AB
+ BC
+ CD
= 2a
+ 2b
+ 2c
= 2(a + b + c)
AQ =
a + b + c
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30. Find in terms of a, b and c: AD AQ MQ NP
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.
Find in terms of a, b and c: AD AQ MQ NP
Deduce a geometric fact about the quadrilateral MNPQ b N b a C M
b + c c
b + c a P c A Q
a + b + c
a + b + c D
MQ = MA
+ AQ
= -a
+ (a + b + c)
= b + c
NP = NC
+ CP
= b
+ c
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31. Find in terms of a, b and c: AD AQ MQ NP
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.
Find in terms of a, b and c: AD AQ MQ NP
Deduce a geometric fact about the quadrilateral MNPQ b N b a C M
b + c c
b + c a P c A Q
a + b + c
a + b + c D
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.
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