2i + 3 j
uur uur uur
1
AB + BC + CD = a + b + 2 (a + b)
© T Madas
A vector is a line with a start and a finish.
It therefore has:
1.
2.
3.
A
A
line of action
a direction
a given size (magnitude)
uuur
B
AB = a
B
uur
BA = - a
© 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
uuur
AB
With a lower case letter which:
is printed in bold
a
or underlined when handwritten
a
In component form if the vector is drawn on a grid:
4
5
© T Madas
uuur
Let AB = a
uuur
CD = 2a
uur
EF = 12 a
uuur
HG = - 2a
© T Madas
B
D
AB = 4
5
A
-5
CD =
4
C
© T Madas
© T Madas
ABCD is a parallelogram
uuur
Let AB = a
uuur
Let AD = b
uuur
DC = a
uuur
BC = b
B
b
a
A
C
a
b
D
© T Madas
ABCD is a parallelogram
uuur
Let AB = a
uuur
Let AD = b
uuur
DC = a
uuur
BC = b
B
b
a
A
C
a
b
D
Now adding vectors
uuur
uuur
uuur
uuur
uuur
AC = AD + DC = b + a = a + b
= AB + BC = a + b
© T Madas
ABCD is a parallelogram
uuur
Let AB = a
uuur
Let AD = b
uuur
DC = a
uuur
BC = b
B
b
a
A
C
a
b
D
Now adding vectors
uuur
uuur
uuur
+ CD = b - a
BD = BC
uur uuur
= BA + AD = - a + b = b - a
© T Madas
© T Madas
ABC is a triangle with M the midpoint of AB and N the
uuur
uuur
midpoint of BC. Let AB = a and AC = b
B
uuur
AM =
1a
2
uuur
M
MB =
N
1a
2
1a
2
1a
2
A
C
b
uuur
uur
uuur
BC = BA + AC = - a + b = b - a
© T Madas
ABC is a triangle with M the midpoint of AB and N the
uuur
uuur
midpoint of BC. Let AB = a and AC = b
B
M
N
1a
2
uuur
MB =
1 b- 1 a
2
2
uuur
BN =
uuur
C
b
uuur
AM =
1 b- 1 a
2
2
1a
2
A
uuur
uur
NC =
1a
2
1a
2
1 b2
1 b2
1a
2
1a
2
uuur
BC = BA + AC = - a + b = b - a
© T Madas
ABC is a triangle with M the midpoint of AB and N the
uuur
uuur
midpoint of BC. Let AB = a and AC = b
B
1b
2
N
1a
2
A
uuur
MB =
1 b- 1 a
2
2
uuur
BN =
uuur
C
b
uuur
AM =
1 b- 1 a
2
2
1a
2
M
uuur
uuur
NC =
1a
2
1a
2
1 b2
1 b2
1a
2
1a
2
uuur
1a
1 b- 1 a = 1 b
+
+
BN
=
MN = MB
2
2
2
2
What is the relationship between AC and MN ?
© T Madas
ABC is a triangle with M the midpoint of AB and N the
uuur
uuur
midpoint of BC. Let AB = a and AC = b
B
1b
2
N
1a
2
uuur
MB =
1 b- 1 a
2
2
uuur
BN =
uuur
A
uur
AM =
1 b- 1 a
2
2
1a
2
M
uuur
uuur
b
C
NC =
1a
2
1a
2
1 b2
1 b2
1a
2
1a
2
uuur
NA = NM + MA = - 12 b - 1 a = - 12 ( b + a) = - 12 (a + b)
uur
NA =
uur
NA =
2
NB + BA = - 12 b + 12 a - a = - 1 b 2
uuur uur
NC + CA = 12 b - 12 a - b = - 1 b 2
uuur
uur
1 a etc etc
2
1 a etc etc
2
© T Madas
© T Madas
ABCDEF is a regular hexagon. M is the midpoint of CE.
uuur
uuur
uuur
AB = a, BC = b and CD = c.
Write and simplify expressions in terms of a, b and c for :
uuur
uuur
uuur
uuur
uuur
uuur
uuur
(a) CE , (b) MD and (c) MB
C
D
c
CE = CD + DE = c - a
a
b
M
B
E
1 c- 1 a
2
2
b
a
A
solution
c
F
uuur
CM = ME = 1 c - 12 a
2
uuur
uuur
uuur
MD = ME + ED
= 12 c - 12 a + a
= 12 c + 12 a
© T Madas
ABCDEF is a regular hexagon. M is the midpoint of CE.
uuur
uuur
uuur
AB = a, BC = b and CD = c.
Write and simplify expressions in terms of a, b and c for :
uuur
uuur
uuur
uuur
uuur
(a) CE , (b) MD and (c) MB
C
D
c
a
b
M
B
E
1 c- 1 a
2
2
b
a
A
solution
c
F
uur
MB = MC + CB
= - 12 c + 12 a - b
= 12 a - 1 c - b
2
= 12 (a - c - 2b )
= 12 (a - 2b - c)
© T Madas
© T Madas
ABCD is a parallelogram. M is the midpoint of AD
uuur
uuur
N is a point of BD so that BN : ND = 2 : 1. AB = a and AD = b.
Write and simplify expressions in terms of a and b for :
uuur
uuur
uuur
uuur
(a) BD , (b) BN (c) MN and (d) NC
B
b
C
solution
a
A
uuur
1b
2
2b- 2 a
3
3
M
uur
uuur
3
3
N
a
D
1b
2
BD = BA + AD = - a + b = b - a
uuur
uuur
BN = 2 BD = 2 b - 2 a
3
© T Madas
ABCD is a parallelogram. M is the midpoint of AD
uuur
uuur
N is a point of BD so that BN : ND = 2 : 1. AB = a and AD = b.
Write and simplify expressions in terms of a and b for :
uuur
uuur
uuur
uuur
(a) BD , (b) BN (c) MN and (d) NC
B
b
C
solution
a
A
uuur
1b
2
uuur
2b- 2 a
3
3
N
M
1b
2
uuur
uuur
a
D
й2 b - 2 aщ
1
+
a
+
MN = MA + AB + BN = - 2 b
кл3
ъ
3 ы
= 1a+ 1b
3
6
© T Madas
ABCD is a parallelogram. M is the midpoint of AD
uuur
uuur
N is a point of BD so that BN : ND = 2 : 1. AB = a and AD = b.
Write and simplify expressions in terms of a and b for :
uuur
uuur
uuur
uuur
(a) BD , (b) BN (c) MN and (d) NC
(e) Using your answers from parts (c) and (d), show that M, N
and C lie on a straight line
B
b
C
solution
a
A
uuur
2b- 2 a
3
3
1b
2
M
uuur
uuur
N
1b
2
a
D
NC = NB + BC = - 23 b + 2 a + b = 23 a + 13 b
3
© T Madas
ABCD is a parallelogram. M is the midpoint of AD
uuur
uuur
N is a point of BD so that BN : ND = 2 : 1. AB = a and AD = b.
Write and simplify expressions in terms of a and b for :
uuur
uuur
uuur
uuur
(a) BD , (b) BN (c) MN and (d) NC
(e) Using your answers from parts (c) and (d), show that M, N
and C lie on a straight line
B
b
C
solution
a
A
2b- 2 a
3
3
1b
2
M
1b
2
6
6
NC = 2 a + 1 b = 1 ( 2a + b )
3
a
3
3
3
6
uuur
NC = 2 a + 1 b
3
MN = 1 a + 1 b = 1 ( 2a + b )
uuur
MN = 1 a + 1 b
N
uuur
3
uuur
3
D
What is the
ratio MN : NC ?
© T Madas
© T Madas
ABCD is a parallelogram. M is the midpoint of AB
uuur
uuur uuur
uuur
N is a point of BD so that BN = 1 BD . AB = 6a and AD = 6b.
3
uuur
(a)
(b)
Find the vector NC in terms of a and b
Prove that MNC is a straight line
C
6b
B
2b - 2a
N
3a
6a
M
3a
A
uuur
6b
uuur
D
uuur
BD = BC + BD = 6b - 6 a
uuur
uuur
BN = 13 BD = 2b - 2a
© T Madas
ABCD is a parallelogram. M is the midpoint of AB
uuur
uuur uuur
uuur
N is a point of BD so that BN = 1 BD . AB = 6a and AD = 6b.
(a)
(b)
3
uuur
Find the vector NC in terms of a and b
Prove that MNC is a straight line
C
6b
B
2b - 2a
N
3a
2a + 4b
M
6a
3a
A
uuur
uuur
6b
D
uuur
NC = NB + BC = - (2b - 2a)+ 6b
= - 2b + 2 a + 6b
= 2a + 4 b
© T Madas
ABCD is a parallelogram. M is the midpoint of AB
uuur
uuur uuur
uuur
N is a point of BD so that BN = 1 BD . AB = 6a and AD = 6b.
(a)
(b)
3
uuur
Find the vector NC in terms of a and b
Prove that MNC is a straight line
C
6b
B
2b - 2a
3a
N
M
a + 2b
2a + 4b
6a
3a
A
uuur
uuur
6b
D
uuur
MN = MB + BN = 3a + (2b - 2a)
= 3a + 2b - 2a
= a + 2b
© T Madas
ABCD is a parallelogram. M is the midpoint of AB
uuur
uuur uuur
uuur
N is a point of BD so that BN = 1 BD . AB = 6a and AD = 6b.
3
uuur
(a)
(b)
Find the vector NC in terms of a and b
Prove that MNC is a straight line
C
6b
B
uuur
MN = a + 2b
2b - 2a
3a
N
M
a + 2b
2a + 4b
uuur
NC = 2a + 4b
6a
uuur
MC = 3a + 6b
3a
A
6b
D
uuur uuur
uuur
MN , NC and MC have the same direction
\ M , N and C are collinear
© T Madas
© 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
c) MQ
d) NP
M
c
a
e) Deduce a geometric fact A
a+b+c Q a+b+c D
about the quadrilateral
P
c
MNPQ
AD = AB + BC + CD = 2a + 2b + 2c = 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
c) MQ
d) NP
M
a
b+c
c
b+c
e) Deduce a geometric fact A
a+b+c Q a+b+c D
about the quadrilateral
P
c
MNPQ
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
c) MQ
d) NP
M
a
b+c
c
b+c
e) Deduce a geometric fact A
a+b+c Q a+b+c D
about the quadrilateral
P
c
MNPQ
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
© T Madas