Announcements
•  Sorry about email trouble – go to Course
Weblink for info, updates
http://astro1.panet.utoledo.edu/~mheben/phys_2130.html
Past, Present, Future
•  Where have we been:
–  Units and Measurements (Ch1, intro to base units, dustoff algebra and math)
–  Motion in One Dimension (Ch2, expression of average
and instantaneous quantities, relationships between a, v,
x, and t, scalars and vectors.
•  Where are we going – Motion in 2 & 3 dimensions
(Ch4)
•  What do we need Now
–  Vectors and Vector Manipulation
Chapter 3: Vectors
•  To describe motions in 2- or 3-dimensions, we
need vectors
•  A vector quantity has both a magnitude and a
direction. e.g., acceleration, velocity, displacement,
force, torque, and momentum.
•  A scalar quantity does not involve a (spatial)
direction. e. g. charge, mass, time, temperature,
energy, etc.
Space
We need three spatial dimensions to describe a definite
location in space. In the Cartesian (rectangular)
coordinate system, the dimensions are labeled: “x”, “y”,
and “z”. All are mutually perpendicular.
y
x
z
Note: z is out of the screen
C2 = A2 + B2
Special Triangles
C
Ø2 A
Ø1
B
Sin Ø1 = cos Ø2 = A/C
cos Ø1 = sin Ø2 = B/C
Tan Ø1 = ctn Ø2 = A/B
SOHCAHTOA
Ø2
2
Ø1
5
1
Ø2
Ø1
4
2
Ø1
3
1 Ø1 = Ø 2= 45o
3
Ø1 = 36.9o
Ø2 = 53.1o
Ø2
Ø2 1
Ø1 = 30o
Ø2 = 60o
Unit Vectors
•  Has a magnitude of 1 and points in a
particular direction. ONLY indicates
DIRECTION!
•  i, j, k, unit vectors in the positive x, y,
z direction, follow right-handed
coordinate system
Many times x, y, z are used instead of i, j, k.
Vector Addition Property
(commutative)
(associative)
Vector in a coordinate system
Magnitude-angle notation: y
a
q
x
a: magnitude
q: relative to +x direction, counter-clockwise is
positive: “clock is negative”
Components of Vectors
•  Component notation vs
magnitude-angle notation
Add vectors by components
Multiplication of Vectors
•  Multiply a vector by a scalar: b = s a
–  Magnitude of b: s times the magnitude of a
–  Direction of b : same as a if s > 0,
opposite of a if s < 0
•  Multiply a vector by a vector
–  Scalar product (tells you the projection of a onto b.)
•  results in a scalar
–  Vector product (tells you the area subtended by a and b.)
•  results in another vector perpendicular to both a and b.
Scalar product
Scalar product of two
vectors a and b
a.b = a b cos f
a, b: magnitude of a, b
f: angle between the
directions of a and b
“dot product”
Maximum if f = 0, 0 if f = 90 degrees
Scalar Product
If a and b are parallel, f = 0o
If a and b perpendicular, f = 90o
Scalar Product
Scalar Product between a and b
y
What is the scalar product
between: a = 3.0 i - 4.0 j and
b
x
a
b = -2.0 i + 2.0 j?
Angle between a and b
a=5
b = 2.828
a.b = (3.0)(-2.0)+(-4.0)(2.0)+(0.0)(0.0) = -6.0 -8.0 = -14.0
cosf = a.b/(a b) = -14.0/14.142 = -0.990
f = 171.9o
y
b
x
a
What is the angle between:
a = 3.0 i - 4.0 j and
b = -2.0 i + 2.0 j?
Vector Product
•  Vector product of two vectors a
and b produce a third vector c
whose magnitude is
c = a b sin f
whose direction follow the right
hand rule
Vector product
If a and b are parallel, = 0o
If a and b perpendicular, Ø = 90o
Check point: Vectors C and D have magnitudes of 3
units and 4 units, respectively. What is the angle
between the directions of C and D if the
magnitude of the vector products C x D is
(a)  Zero?
(b) 12 units?
3-D versus 1-D motion
To describe 3-D motion, we will dissect it into 3
one-dimensional motions. Each 1-D will be a
component in the larger 3-D vector.
All of the concepts from Chapter 2 *will* apply to
each component (x, y, z) in 3-D.
Read Chapter 4 over the weekend!
Chapter 4: Motion in two and three
dimensions
•  Vectors are needed to describe the 2D or 3-D motion
•  Position vector:
r=xi+yj+zk
for example:
r=(3m)i+(2m)j+(4m)k
y
x
z
•  Displacement:
from r1 to r2: D r = r2 - r1
D r = (x2 - x1) i + ( y2 - y1 ) j + ( z2 – z1 ) k
y
x
z
Velocity Vector
•  Average velocity between t1 to t2
•  Instantaneous velocity
v points along the tangent of the path at that position
Acceleration vector
•  Average acceleration between t1 to t2
•  Instantaneous acceleration
Check point:
A)  Can an object accelerate if its speed is
constant?
B) Can an object accelerate if its velocity is
constant?
Example: r = t2 i – ( 2 t +1) j (r in meters and t in seconds)
1) What is the displacement between t = 1s and t = 3s?
2) 
What is the velocity at t = 3 s ?
3)
What is the acceleration at t = 3 s ?
Projectile Motion in 2D
y
v0
q0
0
•  Initial velocity:
v0 = v0x i + v0y j
v0x = v0 cosq0, v0y = v0 sin q0
x
Projectile motion in 2D