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)