The process of vector addition is like following a treasure map.

ARRRR, Ye best learn your vectors!

N

S

W E

Go 10 paces North

N

S

W E

A B

C

D

E

R

This is called the resultant vector.

R = A+B+C+D+E

X

N

S

W E

Add the same vectors in a different order.

B

N

S

W E

The vector sum does not depend on the order of addition.

R=B +A+E+D+C

Vector Addition: Using the "head to tail" graphing method

A vector has components. If the components are on the axes they are called rectangular components. The sum of a vector’s components equals the vector.

X component of A

Y c

om

pon

en

t of

AA

A vector and its components are interchangeable. You can either use the vector or its components, depending on which is easier.

A

Consider this triangle…

HypotenuseOpposite Side

(Ay)

Adjacent Side (Ax)

SOH CAH TOA

Adding Vectors in Real Life…Step 1: Draw a Vector Diagram

A=10.0 m

20.o

B=15 m

30.o

C=10. 0 m

Find The Sum of A + B + C

Adding Vectors in Real Life…

Step 2: Create data table holding x and y components of each vector and the total x and y components of the resultant vector.

A=10.0 m

20.o

B=15

30.o

C=10.0 m

A=10.0 m

20.o

B=15. m

30.o

C=10.0 m

10.0cos20

10.0sin20

15sin30

15cos30

XX

9.397 m 3.420 m7.50 m- 13.0

m0 -10.0 m

X Y

C

B

A

Step 3: Add the vectors along each axis to get the total resultant x and y components.

9.397 m 3.420 m 7.50 m-13.0

m0 -10.0 m

X Y

C

B

A

Total -3.6 m 0.92 m

Remember: When adding you round to the least amount of decimal places (but don’t round until the end!)

0.92 m

3.6 m

Step 4: Draw a Vector Diagram showing only the vector axis sums from step 3.

I dropped the negative sign

because the arrowis pointing in the

negative x direction

0.92 m

3.6 m

R2 =(3.6)2 + (0.92)2

R = 3.7 m

Step 5: Use the Pythagorean Theorem (a2 + b2 = c2) to find the magnitude of the resultant vector.

R = 4 m

3.6 m

R

0.92tan

3.6

14.3

Step 6a: Use a trig function (usually tan) to find the angle.

0.92 m

10o

3.6

R

R = 4 m @ 10o North of West

Step 6b: Specify both magnitude and direction of the vector.

0.92 m

Wow! That’s so much work!

25.0 m

10.0 m 20.o

25cos20o = 23.5

25sin20o = 8.55

X

Example: Add the vectors below.

23.5 m -8.55 m

0 m-10.0 m

X Y

TOTAL

B

A

13.5 m -8.55 m

Example: Add the vectors below.

8.55 m

13.5 m

R

R2 = 13.52 + 8.552

=16.0 m

Tan = 8.55/13.5

= 32.3o

16.0 m,

32.3o south of east.