Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print:

A When dealing with just the

magnitude of a vector in print, an italic letter will be used: A

A

Chapter 4 Vector Addition Equality of Two Vectors

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

Movement of vectors in a diagram Any vector can be moved parallel to

itself without being affected

Chapter 4 Vector Addition Negative Vectors

Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions)

A = -B

Resultant Vector The resultant vector is the sum of a

given set of vectors

Chapter 4 Vector Addition When adding vectors, their

directions must be taken into account

Units must be the same Graphical Methods

Use scale drawings Algebraic Methods

More convenient

Chapter 4 Vector Addition

The resultant is the sum of two or more vectors. Vectors canbe added by moving the tail of one vector to the head of anothervector without changing the magnitude or direction of the vector.

Note: The red vector R has the same magnitude and direction.

Vector Addition

Chapter 4 Vector Addition

Multiplying a vector by a scalar number changes its lengthbut not its direction unless the scalar is negative.

V 2V -V

Chapter 4 Vector Addition

If two vectors are added at right angles, the magnitudecan be found by using the Pythagorean Theorem

R2 = A2 + B 2 and the angle by Tan

OppAdj

If two vectors are added at any other angle, the magnitudecan be found by the Law of Cosines

and the angle by the Law of Sines

Sin Aa

SinB

b

SinCc

cos2222 ABBAR

Chapter 4 Vector Addition

6 meters

8 meters

10 meters

The distance traveled is 14 meters and the displacementis 10 meters at 36º south of east.

62+82=102

6386

tan1

36°

Chapter 4 Vector Addition

A hiker walks 3 km due east, then makes a 30° turn north of east walks another 5 km. What is the distance and displacement of thehiker?

The distance traveled is 3 km + 5 km = 8 km

R2 = 32+52- 2*3*5*Cos 150°R2 = 9+25+26=60R = 7.7 km

5θsin

7.7150sin

193 km

5 km

The displacement is 7.7 km @ 19° north of east

R

Chapter 4 Vector Addition

Add the following vectors and determine the resultant.3.0 m/s, 45 and 5.0 m/s, 135

5.83 m/s, 104

Chapter 4 Vector Addition

Chapter 4 Vector AdditionA boat travels at 30 m/s due east across a river that is 120 m wideand the current is 12 m/s south. What is the velocity of the boatrelative to shore? How long does it take the boat to cross the river? How far downstream will the boat land?

30 m/s 30 m/s

12 m/s12 m/s

The speed will be 22 3012 = 32. 3 m/s @ 21° downstream.

The time to cross the river will be t = d/v = 120 m / 30 m/s = 4 sThe boat will be d = vt = 12 m/s * 4 s = 48 m downstream.

Chapter 4 Vector Addition

Examples

Chapter 4 Vector Addition

Chapter 4 Vector Addition

Chapter 4 Vector Addition

Add the following vectors and determine the resultant.6.0 m/s, 225 + 2.0 m/s, 90 4.80 m/s, 207.9

Chapter 4 Vector Addition

Add the following vectors and determine the resultant.6.0 m/s, 225 + 2.0 m/s, 90

45°2 m6 m

R

•R2 = 22 + 62 – 2*2*6*cos 45•R2 = 4 + 36 –24 cos 45•R2 = 40 – 16.96 = 23•R = 4.8 m

2

sin

8.4

45sin

17

17

R = 4.8 m @ 208

Chapter 4 Vector Addition

A component is a part

It is useful to use rectangular components These are the

projections of the vector along the x- and y-axes

Chapter 4 Vector Addition The x-component of a vector is the

projection along the x-axis

The y-component of a vector is the projection along the y-axis

Then,

cosAAx

sinAAy

yx AA A

Chapter 4 Vector Addition The previous equations are valid only if θ is

measured with respect to the x-axis The components can be positive or negative

and will have the same units as the original vector

The components are the legs of the right triangle whose hypotenuse is A

May still have to find θ with respect to the positive x-axis

x

y12y

2x A

AtanandAAA

Chapter 4 Vector Addition Choose a coordinate system and

sketch the vectors Find the x- and y-components of all

the vectors Add all the x-components

This gives Rx: xx vR

Chapter 4 Vector Addition Add all the y-components

This gives Ry:

Use the Pythagorean Theorem to find the magnitude of the Resultant:

Use the inverse tangent function to find the direction of R:

yy vR

2y

2x RRR

x

y1

R

Rtan

Chapter 4 Vector Addition

Vector components is taking a vector and finding the correspondinghorizontal and vertical components.

sin

cos

AA

AA

y

x

A

Ay

Ax

Vector resolution

Chapter 4 Vector Addition

A plane travels 500 km at 60°south of east. Find the east and south components of its displacement.

500 km

60°

de

ds

de= 500 km *cos 60°= 250 km

ds= 500 km *sin 60°= 433 km

Chapter 4 Vector Addition

Vector equilibrium

Maze Game