VECTOR ADDITION

GraphicalAnalyticalComponent Method

MATH STUFF

What does an ordered pair mean in math? Ex:(2,3)

VECTORS

Quantities having both magnitude and direction Magnitude: How much (think of it as the length of

the line) Direction: Which way is it pointing?

Can be represented by an arrow-tipped line segment

Examples: Velocity Acceleration Displacement Force

QUESTION:

Compare the two vectors. What makes them different

ANSWER

Direction The magnitude or length is exactly the

same

VECTOR TERMINOLOGY

Two or more vectors acting on the same point are said to be concurrent vectors.

The sum of 2 or more vectors is called the resultant (R). A single vector that can replace concurrent

vectors Any vector can be described as having both x

and y components in a coordinate system. The process of breaking a single vector into

its x and y components is called vector resolution.

MORE VECTOR TERMINOLOGY

Vectors are said to be in equilibrium if their sum is equal to zero.

A single vector that can be added to others to produce equilibrium is call the equilibrant (E). Equal to the resultant in magnitude but

opposite in direction.

E + R = 0 E = - R E = 5 N R = 5 N at 180 ° at 0°

WHAT IS THE RESULTANT OF THE FOLLOWING VECTORS?

E= 10 N at 0 degrees R = 20 N at 0 degrees

ANSWER

30 N at 0 degrees

QUESTION

20 N at 45 degrees 10 N at 225 degrees

Do not get freaked out by the angles, Think about it for a second.

ANSWER

10 N at 45 degrees

USING THE GRAPHICAL METHOD OF VECTOR ADDITION:

Vectors are drawn to scale and the resultant is determined using a ruler and protractor.

Vectors are added by drawing the tail of the second vector at the head of the first (tip to tail method). The order of addition does not matter.

The resultant is always drawn from the tail of the first to the head of the last vector.

EXAMPLE PROBLEM

A 50 N force at 0° acts concurrently with a 20 N force at 90°.

R

R and are equal on each diagram.

ADDING VECTORS!

QUESTION: ADD THESE TWO VECTORS TOGETHER

a

b

ANSWER

a

b

R= a+b

MOTION APPLICATIONS Perpendicular vectors act

independently of one another. In problems requesting information

about motion in a certain direction, choose the vector with the same direction.

EXAMPLE PROBLEM:MOTION IN 2 DIMENSIONS

A boat heads east at 8.00 m/s across a river flowing north at 5.00 m/s. What is the resultant velocity of the boat?

A BOAT HEADS EAST AT 8.00 M/S ACROSS A RIVER FLOWING NORTH AT 5.00 M/S.

8.00 m/s E

5.00 m/s N

River width

WHAT IS THE RESULTANT VELOCITY OF THE BOAT?

8.00 m/s E

5.00 m/s N

Draw to scale and measure.

R = 9.43 m/s at 32°

ADVANTAGES AND DISADVANTAGES OF THE GRAPHICAL METHOD

Can add any number of vectors at once

Uses simple tools No mathematical

equations needed

Must be correctly draw to scale and at appropriate angles

Subject to human error

Time consuming

SOLVING VECTORS USING THE ANALYTICAL METHOD

A rough sketch of the vectors is drawn. The resultant is determined using:

Algebra Trigonometry Geometry

QUICK REVIEW

Right Triangle

a

c

b

A

B

C

c is the hypotenuse

c2 = a2 + b2

sin = o/h cos = a/h tan = o/a

A + B + C = 180°

tan A = a/btan B = b/a

B = 180° – (A + 90°)

THESE LAWS WORK FOR ANY TRIANGLE.

a

c

b

C

B A

A + B + C = 180°

Law of sines:

a = b = c sin A sin B sin C

Law of cosines:

c2 = a2 + b2 –2abCos C

USE THE LAW OF: Sines when you

know: 2 angles and an

opposite side 2 sides and an

opposite angle

Cosines when you know: 2 sides and the angle

between them

FOR RIGHT TRIANGLES:

Draw a tip to tail sketch first. To determine the magnitude of the

resultant Use the Pythagorean theorem.

To determine the direction Use the tangent function.

TO ADD MORE THAN TWO VECTORS: Find the resultant for the first two

vectors. Add the resultant to vector 3 and find

the new resultant. Repeat as necessary.

ADVANTAGES AND DISADVANTAGESOF THE ANALYTICAL METHOD

Does not require drawing to scale.

More precise answers are calculated.

Works for any type of triangle if appropriate laws are used.

Can only add 2 vectors at a time.

Must know many mathematical formulas.

Can be quite time consuming.

SOLVING VECTOR PROBLEMS USING THE COMPONENT METHOD

Each vector is replaced by 2 perpendicular vectors called components.

Add the x-components and the y-components to find the x- and y-components of the resultant.

Use the Pythagorean theorem and the tangent function to find the magnitude and direction of the resultant.

VECTOR RESOLUTION

hy = h sin

x = h cos

- + ++

- - +-

x

y

COMPONENTS OF FORCE:

x

y

QUESTION

What are the components of the following force25N @ 12 degrees North of West

ANSWER

West is 180 degrees to 12 degrees north of west is 168 degrees

The X component is -24.45N The Y component is 5.20N You can confirm you answer –X and +Y

would be found in the second quadrant on a graph so this answer makes sense

EXAMPLE:

5 N at 30°

6 N at 135° x y

5 cos 30° = +4.33 5 sin 30° = +2.5

6 cos 45 ° = - 4.24

6 sin 45 ° = + 4.24

+ 0.09 + 6.74

R = (0.09)2 + (6.74)2 = 6.74 N

= arctan 6.74/0.09 = 89.2°

TANGENT FUNCTION

The tangent function has 2 places that it is not defined (you get an error on your calculator) 90 degrees and 270 degrees

The x and y components tell you the angle range

Angle Range Operation

0 to 90 Nothing special needed

90 to 180 Add 180 degrees

180 to 270 Add 180 degrees

270 to 360 Add 360 degrees

QUESTION: CRITICAL THINKING

My X component was negative and my y component was negative as well. My calculator told me that my answer was 22 degrees. What is my true angle?

ANSWER

My evidence: Negative X Negative Y

We are in quadrant three (between 180 degrees and 270)

I got 22 degrees, so I must take 180+22 to get 202 degree as my angle! Using my tangent rules

Solve the following problem using the component method.

10 km at 30

6 km at 120

ADDING VECTORS

Vector X component Y component

6 km @ 120 degrees 6 km * cos(120) = 6 km * sin(120) =

10 km @ 30 degrees 10 km * cos (30) = 10 km * sin(30) =

Add them together

To find the magnitude: pythagorean theorumTo find the direction:

1. Take into account if either X or y is + or –2. Use any trig function SOH CAH TOA to find angle

CRITICAL THINKING QUESTION 2

I get a positive x and a negative y component when I add them together.

What degree range is my angle in?

ANSWER

X is positive so that can only mean either quadrant 1 or 4

Y is negative so that means you have to get quadrant 4 as your answer

270 to 360 degrees

NOTES

Make sure that all angles are measured from the x axis (0 degrees)

Report both the magnitude and the direction otherwise the vector is wrong!

Keep track of signs, They give you a clue to where the angle of the vector actually is.

ADDING TWO VECTORS

Vector X component Y component

12 N @ 135 degrees

15 N @ 200 degrees

Resultant X and Y

Find the resultant magnitude:Find the resultant direction: