Chapter 3 Vectors

Vectors and Scalars

• A Scalar is a physical quantity with magnitude (and units). Examples: Temperature, Pressure, Distance, Speed

• A Vector is a physical quantity with magnitude and direction: Displacement: Washington D.C. is ~ 180 miles N of

Newport News Wind Velocity: 20mi/hr towards SW

Components of a vector, as an alternate to magnitude and direction

• 100 miles 30 degrees north of east, is equivalent to 86.6 miles east followed by 50 miles north

86.6mi

50mi100mi

N

E

Labels for Components of a Vector

• To free ourselves from the points of the compass, we will use x & y instead of E & N

• Vector , magnitude• Components (as ordered pair)

A

AA

),( yx AAA

),(),( yxrrr yx

Trigonometry and Vector Components

• Trigonometry is not a pre-requisite for this course.• Today you will learn ½ of trigonometry, and all

that you need for this course.• In this discussion, we always define the direction

of a vector in terms of an angle counter-clockwise from the + x-axis.

• Negative angles are measured clockwise.

Trigonometry and Circles

• The point P1=(x1,y1) lies on a circle of radius r.

• The line from the origin to P1 makes an angle w.r.t. the x-axis.

• The trigonometric functions sine and cosine are defined by the x- and y-components of P1:

x1

y1r

P1

45-45-90 triangle

• By symmetry, x= y

• Pythagorean Theorem: x

+ y

2 = r2

2· x = r2

x= r/2

• cos(45º) = x /r • cos(45º) = 0.7071• sin(45º) = 1/ 2

30-60-90 Triangle

Vector Addition:Graphical(use bold face for vector symbol)

• A, B, and C are three displacement vectors. Any point can be the origin for a

displacement

• The vector B = 3 paces to E. Notice that B has been translated

from the origin until the tail of B is at the head of A.

• This is the “head-to-tail” method of vector addition.

• Vector addition is commutative, just like ordinary addition: D = A+B+C = C+B+A

Vector Addition, Components

• When we add two vectors, the components add separately: Cx = Ax + Bx

Cy = Ay + By

Velocity Vectors

• Each fish in a school has its own velocity vector.

• If the fish are swimming in unison, the velocity vectors are all (nearly) identical

• We draw each vector at the position of the fish.

Scalar MultiplicationMultiplying a vector by a scalar

• Multiplying a vector by a positive scalar quantity simply re-scales the length (and maybe units) of the vector, without changing direction.

• Multiplying a vector by a negative number reverses the direction of the vector.

x

yA

A

))(6.0( A

Vector Subtraction

• Subtraction is just addition of the additive inverse

1212 vvvv

2112

2112 get to toadd vector to

vvvv

vvvv

x

y

2v

1v

1v

12 vv

Average Velocity Vector

• Net displacement (vector) multiplied by reciprocal of elapsed time (scalar)

1212

12

12

1 )(

ttrr

t

r

tt

rrvav

r1

r2

A whale comes to the surface to breathe, and then dives at an angle 20.0° below the horizontal. Answer the following questions if the whale continues in a straight line for 140 m. (a) How deep is it? (b) How far has it traveled horizontally?

Example 1

Example 2

Consider the vectors in Figure 3-36, in which the magnitudes of A, B, C, and D are respectively given by 15 m, 20 m, 10 m, and 15 m. Express the sum, A + C + D, in unit vector notation.

Relative Motion

pg pt tgV V V ������������������������������������������

Relative Motion

pg pt tgV V V ������������������������������������������

Relative Motion Example

As an airplane taxies on the runway with a speed of 15.4 m/s, a flight attendant walks toward the tail of the plane with a speed of 1.30 m/s. What is the flight attendant's speed relative to the ground?