Transcript Walker, Chapter 3

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
N
50mi
86.6mi
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 A , magnitude A  A

• Components (as ordered pair) A  ( Ax , Ay )

r  (rx , ry )  ( x, y )
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 q1 w.r.t. the xaxis.
• The trigonometric functions
sine and cosine are defined by
the x- and y-components of P1:
P1
r
y1
q
x1
45-45-90 triangle
• By symmetry,
 x1 = y1
• Pythagorean Theorem:
 x12 + y12 = r2
 2· x12 = r2
 x1 = r/2
• cos(45º) = x1 /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 Multiplication
Multiplying 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.
A
y

A

(0.6)( A)
x
Vector Subtraction
• Subtraction is just addition of the additive inverse
  

v2  v1  v2   v1
 


v2  v1  vector to add to v1 to get v2
 
 
y 
v2  v1  v1  v2
v2
 
v2  v1

 v1

v1
x
Average Velocity Vector
• Net displacement
(vector) multiplied by
reciprocal of elapsed
time (scalar)
 

r2  r1 r

vav 

t2  t1 t

  1 
 r2  (r1) 

t

t
 2 1
r1
r2
Example 1
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 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
Vpg  Vpt  Vtg
Relative Motion
Vpg  Vpt  Vtg
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?