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DESCRIPTION
Introduction to vectors
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Vectors
By Aruna
A VECTOR?
□Describes the motion of an object□A Vector comprises
□Direction□Magnitude
□We will consider□Column Vectors□General Vectors□Vector Geometry
Size
Column Vectors
a
Vector a
COLUMN Vector
4 RIGHT
2 up
NOTE!
Label is in BOLD.
When handwritten, draw a wavy line under the label
i.e. ~a
2
4
Column Vectors
b
Vector b
COLUMN Vector?
3
2
3 LEFT
2 up
Column Vectors
n
Vector u
COLUMN Vector?
4
2
4 LEFT
2 down
Describe these vectors
b
a
c
d
2
3
1
3
4
1
4
3
Alternative labelling
CD++++++++++++++
EF++++++++++++++
AB
A
B
C
DF
E
G
H
GH++++++++++++++
General VectorsA Vector has BOTH a Length & a Direction
k can be in any position
k
k
k
k
All 4 Vectors here are EQUAL in Length andTravel in SAME Direction.All called k
General Vectors
kA
B
C
D
-k
2k
F
E
Line CD is Parallel to AB
CD is TWICE length of AB
Line EF is Parallel to AB
EF is equal in length to AB
EF is opposite direction to AB
Write these Vectors in terms of k
k
A
B
C
D
E
F G
H
2k1½k ½k
-2k
Combining Column Vectors
AB
AB
k
A
B
C
D3k
++++++++++++++AB
1
2k
231
++++++++++++++AB
6
3
++++++++++++++AB
2k++++++++++++++CD
221
++++++++++++++CD
4
2
++++++++++++++CD
A
B
C
Simple combinations
1
4AB
5AC =
4
++++++++++++++
3
1BC
db
ca
d
c
b
a
Vector Geometry
OP a++++++++++++++
OR b++++++++++++++
RQ++++++++++++++Consider this parallelogram
Q
O
P
Ra
b
PQ++++++++++++++
Opposite sides are Parallel
OQ OP PQ++++++++++++++++++++++++++++++++++++++++++
OQ OR RQ++++++++++++++++++++++++++++++++++++++++++
OQ is known as the resultant of a and b
a+b
b+ a
a+b b+ a
Resultant of Two Vectors
□Is the same, no matter which route is followed
□Use this to find vectors in geometrical figures
Example
Q
O
P
Ra
b
.SS is the Midpoint of PQ.
Work out the vector OS
PQOPOS ½
= a + ½b
Alternatively
Q
O
P
Ra
b
.SS is the Midpoint of PQ.
Work out the vector OS
OS OR RQ QS ++++++++++++++++++++++++++++++++++++++++++++++++++++++++
= a + ½b
= b + a - ½b
= ½b + a
Example
AB
C
p
q
M M is the Midpoint of BC
Find BC
AC= p, AB = q
BC BA AC= += -q + p
= p - q
Example
AB
C
p
q
M M is the Midpoint of BC
Find BM
AC= p, AB = q
BM ½BC=
= ½(p – q)
Example
AB
C
p
q
M M is the Midpoint of BC
Find AM
AC= p, AB = q
= q + ½(p – q)
AM + ½BC= AB
= q +½p - ½q
= ½q +½p = ½(q + p) = ½(p + q)
Alternatively
AB
C
p
q
M M is the Midpoint of BC
Find AM
AC= p, AB = q
= p + ½(q – p)
AM + ½CB= AC
= p +½q - ½p
= ½p +½q = ½(p + q)
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