Vectors
Vectors
A VECTOR? • Describes the motion of an object • A Vector comprises – Direction – Magnitude
Size
• We will consider – Column Vectors – General Vectors – Vector Geometry
Column Vectors Vector a a ~
NOTE! Label is in BOLD.
a
4 2
2 up
4 RIGHT
When handwritten, draw a straight line under the label i.e.
COLUMN Vector
a
Column Vectors Vector b 2 up
b
3 LEFT
3 2 COLUMN Vector?
Column Vectors Vector u 2 down
n 4 LEFT
4 2 COLUMN Vector?
Describe these vectors 4 1 1 a 3
b
c
2 3 d
4 3
Alternative labelling B
D
EF
E
AB
F
CD G C
A
GH H
General Vectors A Vector has BOTH a Length & a Direction
All 4 Vectors here are EQUAL in Length and Travel in SAME Direction. All called k
k k k
k k can be in any position
General Vectors Line CD is Parallel to AB
B A
CD is TWICE length of AB
k
D 2k
Line EF is Parallel to AB
E
C
-k F
EF is equal in length to AB EF is opposite direction to AB
Write these Vectors in terms of k B
k
D 2k
½k
1½k
F
G
E A
C
-2k H
Combining Column Vectors B AB
A
k
C AB
D
2 k 1 CB D 32kk A 2 A B 3 2 C D 2 1 1 64 AB C D 32
Simple combinations C
4 AB 1 BC
1 3
5 AC = 4
B A a b
c a c d b d
Vector Geometry Consider this parallelogram Q
P R
a b
OP a
RQ
OR b
PQ
Opposite sides are Parallel
OQ OP PQ
a + b
OQ OR RQ
b + a
O
a + b b + a
OQ is known as the resultant of a and b
Resultant of Two Vectors • Is the same, no matter which route is followed • Use this to find vectors in geometrical figures
Example S is the Midpoint of PQ.
.
Q
S
P
OS OP ½ PQ R
a b O
Work out the vector OS
= a + ½b
Alternatively S is the Midpoint of PQ.
.
Q
S
P
OS OR RQ QS
R
a b O
Work out the vector OS
= b + a - ½b = ½b + a = a + ½b
Example C AC= p, AB = q p
A
M
q
Find BC
M is the Midpoint of BC
B
BC = BA + AC = -q + p =p-q
Example C AC= p, AB = q p
A
M
q
Find BM
BM = ½BC
= ½(p – q)
M is the Midpoint of BC
B
Example C AC= p, AB = q p
A
M is the Midpoint of BC
M
q
Find AM
AM = AB
B
+ ½BC
= q + ½(p – q) = q +½p - ½q = ½q +½p
= ½(q + p)
= ½(p + q)
Alternatively C AC= p, AB = q p
A
M is the Midpoint of BC
M
q
Find AM
B
AM = AC + ½CB
= p + ½(q – p) = p +½q - ½p = ½p +½q
= ½(p + q)
A VECTOR? • Describes the motion of an object • A Vector comprises – Direction – Magnitude
Size
• We will consider – Column Vectors – General Vectors – Vector Geometry
Column Vectors Vector a a ~
NOTE! Label is in BOLD.
a
4 2
2 up
4 RIGHT
When handwritten, draw a straight line under the label i.e.
COLUMN Vector
a
Column Vectors Vector b 2 up
b
3 LEFT
3 2 COLUMN Vector?
Column Vectors Vector u 2 down
n 4 LEFT
4 2 COLUMN Vector?
Describe these vectors 4 1 1 a 3
b
c
2 3 d
4 3
Alternative labelling B
D
EF
E
AB
F
CD G C
A
GH H
General Vectors A Vector has BOTH a Length & a Direction
All 4 Vectors here are EQUAL in Length and Travel in SAME Direction. All called k
k k k
k k can be in any position
General Vectors Line CD is Parallel to AB
B A
CD is TWICE length of AB
k
D 2k
Line EF is Parallel to AB
E
C
-k F
EF is equal in length to AB EF is opposite direction to AB
Write these Vectors in terms of k B
k
D 2k
½k
1½k
F
G
E A
C
-2k H
Combining Column Vectors B AB
A
k
C AB
D
2 k 1 CB D 32kk A 2 A B 3 2 C D 2 1 1 64 AB C D 32
Simple combinations C
4 AB 1 BC
1 3
5 AC = 4
B A a b
c a c d b d
Vector Geometry Consider this parallelogram Q
P R
a b
OP a
RQ
OR b
PQ
Opposite sides are Parallel
OQ OP PQ
a + b
OQ OR RQ
b + a
O
a + b b + a
OQ is known as the resultant of a and b
Resultant of Two Vectors • Is the same, no matter which route is followed • Use this to find vectors in geometrical figures
Example S is the Midpoint of PQ.
.
Q
S
P
OS OP ½ PQ R
a b O
Work out the vector OS
= a + ½b
Alternatively S is the Midpoint of PQ.
.
Q
S
P
OS OR RQ QS
R
a b O
Work out the vector OS
= b + a - ½b = ½b + a = a + ½b
Example C AC= p, AB = q p
A
M
q
Find BC
M is the Midpoint of BC
B
BC = BA + AC = -q + p =p-q
Example C AC= p, AB = q p
A
M
q
Find BM
BM = ½BC
= ½(p – q)
M is the Midpoint of BC
B
Example C AC= p, AB = q p
A
M is the Midpoint of BC
M
q
Find AM
AM = AB
B
+ ½BC
= q + ½(p – q) = q +½p - ½q = ½q +½p
= ½(q + p)
= ½(p + q)
Alternatively C AC= p, AB = q p
A
M is the Midpoint of BC
M
q
Find AM
B
AM = AC + ½CB
= p + ½(q – p) = p +½q - ½p = ½p +½q
= ½(p + q)
