GCSE: Vectors

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# GCSE: Vectors - PowerPoint PPT Presentation

GCSE: Vectors. Dr J Frost (jfrost@tiffin.kingston.sch.uk). Last modified: 26 th January 2014. Starter. b. ( ). ( ). ( ). ( ). ( ). ( ). -1 1. -4 -4. -4 2. 5 1. 2 -3. 1 3. ?. ?. c =. a =. d =. e =. f =. b =. a. d. ?. ?. f. e. ?. ?. c.

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## GCSE: Vectors

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### GCSE: Vectors

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Starter

b

( )

( )

( )

( )

( )

( )

-1

1

-4

-4

-4

2

5

1

2

-3

1

3

?

?

c =

a =

d =

e =

f =

b =

a

d

?

?

f

e

?

?

c

• Bro Tip: Ensure you can distinguish between coordinates and vectors.
• Coordinates represent positions.
• Vectors represent movement.

What is a vector?

A vector is an entity with 2 properties:

Direction

Magnitude (the length)

?

?

Vectors are equal if:

Same direction & magnitude

Vectors are parallel if:

Same/opposite direction

?

?

Writing Vectors

Just like conventional algebra, we can represent vectors as variables.

Y

There’s 3 ways in which can represent the vector from point X to Z:

(in bold)

(with an ‘underbar’)

b

Z

a

X

Determine:

XY =

a

+ b

?

XZ = 2a

?

XR = 2a + 2b

Click to Start Bromanimation

T

R

XQ = 2a + b

?

M

3

2

XM = a + b

?

Y

Q

YZ = a – b

?

b

b

b

b

b

b

RS = -2b – a

?

a

a

a

a

a

a

1

2

X

S

Z

?

MQ = - b+a

(M is the midpoint of the line YT)

Parallel or not parallel?

We earlier said that vectors are parallel if they have the same direction (but could have different magnitudes).

No



Yes



No

Yes

No



Yes



No

Yes

For ones which are parallel, show it diagrammatically.

Therefore parallel (in the context of scalars) if:

we can multiply one vector by a scalar to get the other.

?

ζ

Dr Frost

GCSE – Vectors – Lesson 2

Objectives: Be able to further manipulate vectors, particular when considering portions of vectors.

Example 1

is the midpoint of and the midpoint of .

and

Determine:

?

?

?

?

?

Bro Tip: For more complicated vectors, express it as a sum of other vectors.

?

Quick fire ratio

P

A

B

z

The vector

Find the following vectors given the specified ratios:

?

?

?

?

?

?

(use the front for blue)

A

B

C

D

Given that M is the midpoint of BC, determine AM.

B

6a

M

A

4b

C

3a + 2b

3a + b

2a + 3b

a + b

Given that M is the midpoint of BC and Q is the point such that AQ:QB = 3:1, determine MQ.

B

Q

6a

M

A

4b

C

1.5a – 3b

a – 2b

1.5a – 2b

a – 3b

Exercises

Exercise 2 on sheet.

ζ

Dr Frost

GCSE – Vectors – Lesson 3

A challenging one!

The ratio of the lengths OM to MQ is 3:2.

The ratio of the lengths PN to NR is 4:1.

Find MN

P

a

N

R

O

b

c

?

M

Q

Recap

Two vectors are parallel if:

One is a multiple of the other.

?

Points A, B and C form a straight line if:

and are parallel (and B is a common point).

?

C

B

A

GCSE Mark Scheme

1 mark

1 mark

1 mark

Simplifying them to and

Write this down!

1 mark

“NM is a multiple of MC”

(+ they have a common point M)

Why do you think this is significant?

GCSE Mark Scheme

1 mark

2a + 0.4(3b – 2a) or 3b + 0.6(2a – 3b)

1 mark

Simplified to 1.2a + 1.2b

1 mark

Explicitly stating that “1.2(a + b) is parallel to a + b”

You need to explicitly state the conclusion!

C

B

A

The approach to prove that ABC is a straight line is to:

Prove that AB is parallel to BC.

We can show that these are parallel by stating:

“Vector BC is a multiple of AB”.

?

?