Transcript Addition of vectors (i) Triangle Rule [For vectors with a common point] C AB  BC  AC B A.

Addition of vectors
(i) Triangle Rule [For vectors with a common point]
C
AB  BC  AC
B
A
(ii) Parallelogram Rule [for vectors with same initial point]
D
C
AB  AC  AD
B
A
(iii) Extensions follow to three or more vectors
r
p+q+r
q
p
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.
A few examples
b–a

a
b
a
b
a) Which vector is represented by p + q ?
b) Which vector is represented by p – q ?
q
p
p
p+q
p
q
p–q
–q
B
CB = CA + AB
= - AC + AB
= AB – AC
C
A
Position Vectors
Relative to a fixed point O [origin] the position of a Point P in space
is uniquely determined by OP
P
p
O
OP is a position vector of a point P.
We usually associate p with OP
A very Important result!
B
AB = b - a
b
A
a
O
The Midpoint of AB
A
M
OM = ½(b + a)
a
B
b
O
Vectors Questions
P
Example
In the diagram, OA = AP and BQ = 3OB.
N is the midpoint of PQ.
OA  a and OB  b
A
a
N
Express each of the following
vectors in terms of a, b or a and
b.
O
b
B
(a)
(e)
AB
PQ
AP
(f)
PN
(b)
(c)
OQ (d) PO
(g)
(h)
ON
AN
Q
OA = AP and BQ = 3OB
a)
AB 
b)
AP 
c)
OQ 
d)
PO 
ab
P
a
A
a
4b
N
O
 2a
e)
PQ   2a  4b
f)
PN 
b
B
 a  2b
g) ON  OP  PN
 2a  a  2b
 a  2b
Q
h)
AN 
AP  PN
 a  a  2b
 2b
Example
AB representing the vector m
and AF , representing the vector n. Find the vector representing AD
ABCDEF is a regular hexagon with ,
m
A
n
B
m+n
m
F
C
E
AD  2m  2n
D
Example
M, N, P and Q are the mid-points of OA, OB, AC and BC.
B
a) (i) BC = BO + OC
OA = a, OB = b, OC = c
= c–b
A
(ii) NQ = NB + BQ
N
M
a
Q
b
= c
P
(ii) MP = MA + AP
O
c
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?
= c
MNPQ is a parallelogram as NQ and MP are equal and parallel.
Example
In the diagram, OA  a and
has ratio 1 : 2.
OB  b , OC = CA, OB = BE and BD : DA
a) Express in terms of a and b
(i)
A
BA
(ii)
BD
(iii)CD
(iv) CE
C
D
b) Explain why points C, D
and E lie on a straight line.
a
O
b
B
E
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 BA ,
is parallel to
DE
CA, 2BC