Vectors
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Vectors. A VECTOR?. Describes the motion of an object A Vector comprises Direction Magnitude We will consider Column Vectors General Vectors Vector Geometry. Size. NOTE! Label is in BOLD . When handwritten, draw a wavy line under the label i.e. a. Column Vectors. Vector a.

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Vectors

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Vectors

Vectors


A vector

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

NOTE!

Label is in BOLD.

When handwritten,

draw a wavy line

under the label

i.e.

a

Column Vectors

Vector a

2 up

4 RIGHT

COLUMN Vector


Column vectors1

b

Column Vectors

Vector b

2 up

3 LEFT

COLUMN Vector?


Column vectors2

n

Column Vectors

Vector u

2 down

4 LEFT

COLUMN Vector?


Describe these vectors

b

a

d

c

Describe these vectors


Alternative labelling

Alternative labelling

F

B

D

E

G

C

A

H


General vectors

k

k

k

k

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 can be in any position


General vectors1

k

General Vectors

Line CD is Parallel to AB

B

CD is TWICE length of AB

D

A

2k

Line EF is Parallel to AB

E

EF is equal in length to AB

C

-k

EF is opposite direction to AB

F


Write these vectors in terms of k

k

Write these Vectors in terms of k

B

D

2k

F

G

½k

1½k

E

C

-2k

A

H


Combining column vectors

B

k

D

A

C

Combining Column Vectors


Simple combinations

C

B

A

Simple combinations


Vector geometry

Q

P

R

a

b

O

Vector Geometry

Consider this parallelogram

Opposite sides are Parallel

OQ is known as the resultant of a and b


Resultant of two vectors

Resultant of Two Vectors

  • Is the same, no matter which route is followed

  • Use this to find vectors in geometrical figures


Example

.

Q

S

S is the Midpoint of PQ.

Work out the vector

P

R

a

b

O

Example

= a + ½b


Alternatively

.

Q

S

S is the Midpoint of PQ.

Work out the vector

P

R

a

b

O

Alternatively

- ½b

= b

+ a

= ½b + a

= a + ½b


Example1

C

p

M

Find BC

=

+

A

q

B

BC

BA

AC

AC= p, AB = q

Example

M is the Midpoint of BC

= -q + p

= p - q


Example2

C

p

M

Find BM

=

½BC

A

q

B

BM

AC= p, AB = q

Example

M is the Midpoint of BC

= ½(p – q)


Example3

C

p

M

Find AM

+ ½BC

=

A

q

B

AM

AB

AC= p, AB = q

Example

M is the Midpoint of BC

= q + ½(p – q)

= q +½p - ½q

= ½q +½p

= ½(q + p)

= ½(p + q)


Alternatively1

C

p

M

Find AM

+ ½CB

=

A

q

B

AC

AM

AC= p, AB = q

Alternatively

M is the Midpoint of BC

= p + ½(q – p)

= p +½q - ½p

= ½p +½q

= ½(p + q)


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