Vectors in Two & Three Dimensions

8/17/2019 Vectors in Two & Three Dimensions

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AQA Core 4 Vectors

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Section 1: Vectors in two and three dimensions

Notes and Examples

These notes contain subsections on

Writing a vector in component form Vectors in three dimensons

Multiplying a vector by a scalar

Adding and subtracting vectors

Equal vectors and position vectors

Unit vectors

Writing a vector in component form

A vector quantity has both magnitude (size) and direction. A scalar quantity has magnitude only.

Vectors may be written in bold, a, or underlined, a, or with an arrow above, a .Two vectors are equal if they have the same magnitude and direction.

Vectors are often expressed in the forma

b

, where a is the number of units

in the x-direction, and b is the number of units in the y-direction.

The vector can also be written using the unit vectors i and j.

i is a unit vector in the x direction. j is a unit vector in the y direction.

The magnitude of a vector given in component form is found usingPythagoras’s theorem.

So the vectora

b

c has magnitude:

2 2a b c

This vector is3

4

or 3i + 4 j

A unit vector hasa magnitude of 1.

j

i

The magnitude of avector is sometimescalled the modulus.



AQA C4 Vectors 1 Notes and Examples

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Vectors in three dimensions

3-D coordinates can be plotted on a grid like this one:

The length of the vector 1 2 3a a a a i j k is:

2 2 2

1 2 3a a a a

Example 1

The points A and B have coordinates (4, -1, 2) and (-1, 3, 1) respectively.

Find AB .

Solution

4

OA 1

2

and

1

OB 3

1

AB OB OA

1 4 5

AB 3 1 4

1 2 1

2 2 2AB ( 5) 4 ( 1) 25 16 1 42

For further practice in examples like the ones above, use the interactivequestions Magnitude of a vector .

Multiplying a vector by a scalar

To multiply a vector by a scalar (number) simply multiply each of thecomponents by the scalar.

The coordinates are always

given in the order ( x, y, z ).

P(2, 5, 6)

y

z

x




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Note:

when the scalar is positive the direction of the vector remains thesame but the length (or magnitude) of the vector increases by thesame factor.

when the scalar is negative the direction of the vector is reversed andagain the length (or magnitude) of the vector increase.

Example 2

2

3

a

(i) Find 4a

(ii) Find the value of a

(iii) Write down the value of 4a

Solution

(i)2 8

4 43 12

a

(ii) 2 22 ( 3) 4 9 13 a

(iii) 4 4 4 13 a a

The same principle applies when working with vectors in three dimensions.

Example 3

5

7

a

Find a .

Solution5

7

a , so

5

7

a

Adding and subtracting vectors

To add/subtract vectors simply multiply add/subtract the i components andthen the j components. Adding two or more vectors is called finding the resultant.

This is the same as multiplying by -1.

Just reverse the signs!




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Example 4

(i) Find the resultant of5 3

and7 2

.

(ii) Work out9 5

8 3

Solution

(i) To find the resultant you need to add the vectors.

5 3 2

7 2 5

You can see this more clearly in this diagram:

(ii)9 5 4

8 3 5

Again, the same ideas can be applied to vectors in three dimensions.

The Geogebra resource Add ing and subtract ing vectors demonstrates thegeometrical interpretation of vector addition and subtraction. You may also

find the Mathcentre video Introdu ct ion to vectors useful.

For further practice in examples like the ones above, use the interactivequestions Operat ions on vectors .

Equal vectors and position vectors

Two vectors are equal if they have the same magnitude and direction.

They do not have to be in the same place!

Example 5

The diagram shows a parallelogram ABCD.

5i

-7 j

-3i

2 j

-5 j

2i The resultant is shownby a double headed

arrow.

A B

CD

E




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DA a , AE b , AB c

(a) Find in terms of a, b and c the vectors:

(i) CB (ii) BC (iii) BD .

(b) Find two equivalent expressions for AC .

Solution

(a) (i) CB DA a

(ii) BC CB a

(iii) BD BA AD

BD c a

(b) AC AB BC

AC c a

Also AC 2AE 2 b

A position vector is a vector which starts at the origin.So if two position vectors are equal they will be in the same place!

For example the point A (5, -3) has the position vector OA 5 3 i j .

You need to know that

AO OA

AB OA OB

So AB OB OA

The mid-point, M, has position vector:12

OM OA AB

You can see the reason for these results more clearly in this diagram:

Example 6The points A and B have coordinates (2, 4) and (5, -1) respectively.

(i) Write down the position vectors OA and OB .

(ii) Find the vector AB .

O

A

B

x

y

OB

AO OA

M AB

OM




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(iii) Find the position vector of the mid-point, M of AB.

Solution

(i) OA =2

4

OB =5

1

(ii) 5 2 3

AB OB OA1 4 5

(iii) 1

2

12

1

2

1

2

1

2

OM OA AB

2 31

4 52

12

24

3

1

Unit vectors

A unit vector has a magnitude of 1.

i and j are examples of unit vectors.

You need to be able to find a unit vector which has the same direction as agiven vector, a.You do this by:

Finding the magnitude of the vector, a

Dividing a by its magnitude, a

The unit vector of a is written a

Example 7

Find the unit vector in the direction of2

3

a

Solution2 22 ( 3) 4 9 13 a

2

13

3

13

a

Say ‘a hat’.

Vectors in Two & Three Dimensions

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