Vector Components

Addition of multiple vectors via the graphical solution is complex

A B C D E

E

A

DC

B

It’s easier to use the mathematical method that will be described now

Mathematical MethodAll vectors are placed on the origin of a

Cartesian Coordinates System. Each Vector is replaced with it components

Mathematical MethodAll vectors are placed on the origin of a

Cartesian Coordinates System. Each Vector is replaced with it components

X

Y

AD

C

B

The vector components of vector A are two perpendicular vectors Ax and Ay that are

parallel to X and Y axis respectively, add their resultant vector is equal to A.

x

y

A

xA��������������

yA��������������

x yA A A ������������������������������������������

Calculation of vector components

22 )()(YXAAA

X

Y

AA

tg

cosAAX

X

YA

θ

sinAAY

xA��������������

yA��������������

PythagorasVectorModule

Example

Calculate vector A components

x

y30A

060

30 60 15cos cosXA A

30 60 15 3sin sinYA A

xA��������������

yA��������������

Vector representation depend on the chosen coordinate system (CS).

Different CS does not change the vector,but only it’s representation

x

y

x’

y’

ax

ay

ax’ay’

The vector magnitude is independent of the CS, hence we have freedom to select the CS to ease

the calculation

Vector addition, is done by adding the respective components

Sum of the respective components yield the resultant vector components,

hence we can find the resultant vector

X

Y

A

θ

xA��������������

yA�������������� β

B

C

C A B

xB

��������������yB

YYY

XXX

BAC

BAC

BAC

22

YXCCC

X

Y

A

θ

xA��������������

yA�������������� β

B

C

xB

��������������yB

��������������yC

��������������XC

Y

X

Ctg

C

Unit Vector

The unit vector is marked with a ^ signabove the letter

X

Unit vector is a vector the has a magnitude of 1 with no units, and has a direction

It is described by it’s components as any vector.

The CS unit vectors are unit vectors pointing parallel to the X,Y and Z axes: x, y, z

x

y

z

x

y

z

Representing a vector by it’s components is as follows:

ˆ ˆ ˆ ��������������

X y zA A x A y A z

Determine unit vector components of A

ˆ ˆ ˆ8 6 10A x y z

Draw the vector on the CS

x

y

z

8x

6 ˆ y

10z

A

ˆ ˆ ˆ8 6 10A x y z

Magnitude of A

x

y

z

8x

6 ˆ y

10z

2 2 28 6 10 14.14A A

ˆ ˆ ˆ8 6 10ˆ ˆ ˆ ˆ0.56 0.42 0.714.14

A x y zA x y z

A

Divide each component by the vector magnitude, results in a unit vector having same direction a vector A

x

y

z

8x

6 ˆ y

10zA

x

y

3 cm

A

B6 cm

450

Calculate:1.The components of vectors A and B2.The resultant vector components3.The resultant vector magnitude and direction4.Find the unit vector parallel to the resultant

vector5.A vector with magnitude 10 that is parallel to

the resultant vector

3 cmx

y

A

B 6 cm

450

The components of vectors A and B

0 0

3 0

6 45 6 45 4 2 4 2

ˆ ˆ

cos ˆ sin ˆ . ˆ . ˆ

A x y

B x y x y

3 cmx

y

A

B 6 cm

450

The resultant vector components

3 0

4 2 4 2

ˆ ˆ

. ˆ . ˆ

A x y

B x y

3 4 2 0 4 2

1 2 4 2

( . ) ˆ ( . )ˆ

. ˆ . ˆ

C x y

C x y

C A B

The resultant vector magnitude and direction

x

y

2 2

0

1 2 4 2 4 37

4 23 5

1 2

74 05

. . .

..

.

.

C cm

tg

4.37

-1.2

740

4.2

Above Negative X Axis

C= 1 2 4 2. ˆ . ˆC x y

x

y

4.37

-1.2

740

4.2C= 1 2 4 2. ˆ . ˆC x y

Find the unit vector parallel to the resultant vector

1 2 4 20 27 0 96

4 37

. ˆ . ˆˆ . ˆ . ˆ.

C x yC x y

C

A vector with magnitude 10 that is parallel to the resultant vector

0 27 0 96ˆ . ˆ . ˆC x y

10 10 0 27 0 96 2 7 9 6ˆ ( . ˆ . ˆ) . ˆ . ˆD C x y x y

Unit vector parallel to resultant vector