Vector Addition

GraphicalAnalytical

Component Method

Vectors Quantities having both magnitude and

direction Can be represented by an arrow-tipped

line segment Examples:

Velocity Acceleration Displacement Force

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°

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.

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. If the river is 80.0 m wide, how long

will the boat take to cross the river? How far down stream will the boat be

carried in this amount of time? 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

80.0 m

If the river is 80.0 m wide, how long will the boat take to cross the river?

v = d / t

t = d / v

80.0 m8.00 m/s

= 10.0 s

How far down stream will the boat be carried in this amount of time?

v = d / t

d = vt

5.00 m/s (10.0 s) = 50.0 m

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 ReviewRight 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 Disadvantages

of 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

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°

Advantages of the Component Method:

Can be used for any number of vectors.

All vectors are added at one time.

Only a limited number of mathematical equations must be used.

Least time consuming method for multiple vectors.

Solve the following problem using the component method.

10 km at 30

6 km at 120