Transcript Mathematical Concepts: Polynomials, Trigonometry and Vectors

Mathematical Concepts:
Polynomials, Trigonometry
and Vectors
AP Physics C
20 Aug 2009
Polynomials review





“zero order”
“linear”:
“quadratic”:
And so on….
Inverse functions

Inverse

Inverse square
f(x) = mx0
f(x) = mx1 +b
f(x) = mx2 + nx1 + b
f x  
a
x
a
f x   2
x
Polynomial graphs
Linear
Quadratic
Inverse
Inverse
Square
Right triangle trig

Trigonometry is merely definitions and
relationships.

Starts with the right triangle.
c
a

b
a
sin  
c
b
cos  
c
a
tan  
b
Special Right Triangles



30-60-90 triangles
45-45-90 triangles
37-53-90 triangles (3-4-5 triangles)
Trigonometric
functions & identities
Trig functions
Reciprocal trig
functions
Reciprocal trig
functions
1
csc x 
sin x
1
sec x 
cos x
1
cot x 
tan x
sin x
cos x
tan x
Trig identities
1  sin x  cos x
2
2
sin x
tan x 
cos x
y  sin x  x  sin 1 y
y  cos x  x  cos 1 y
y  tan x  x  cot 1 y
Vectors

A vector is a quantity that has both a
direction and a scalar


Force, velocity, acceleration, momentum,
impulse, displacement, torque, ….
A scalar is a quanitiy that has only a
magnitude

Mass, distance, speed, energy, ….
Cartesian coordinate system

r  x a x  y a y  za z
or

r  x a x  y a y  za z

r  xa i  ya j  za k
Resolving a 2-d vector

“Unresolved” vectors are given by a
magnitude and an angle from some
reference point.


Break the vector up into components by
creating a right triangle.
The magnitude is the length of the
hypotenuse of the triangle.
Resolving a 2-d vector
(example #1)

A projectile is launched from the
ground at an angle of 30 degrees
traveling at a speed of 500 m/s.
Resolve the velocity vector into x and y
components.
Vector addition
graphical method
+
=
+
=
Vector addition
numerical method

Add each component of the vector
separately.



The sum is the value of the vector in a
particular direction.
Subtracting vectors?
To get the vector into “magnitude and
angle” format, reverse the process
Vector addition example #1
Three contestants of a game show are brought to the
center of a large, flat field. Each is given a compass, a
shovel, a meter stick, and the following directions:
72.4 m, 32 E of N
57.3 m, 36 S of W
17.4 m, S
The three displacements are the directions to where
the keys to a new Porche are buried. Two contestants
start measuring, but the winner first calculates where to
go. Why? What is the result of her calculation?
Vector Multiplication
Dot Product


The dot product (or
scalar product), is
denoted by:
It is the projection
of vector A
multiplied by the
magnitude of
vector B.
   
A  B  A B cos 
Vector multiplication
Dot product

In terms of components, the dot product
can be determined by the following:
 
A  B  Ax Bx  Ay By  Az Bz
Vector multiplication
Dot product Example #1
Find the scalar product of the following
two vectors. A has a magnitude of 4,
B has a magnitude of 5.

B
A
50º
53º
Vector Multiplication
Dot Product Example #2

Find the angle between the two vectors

A  2iˆ  3 ˆj  kˆ
B  4iˆ  2 ˆj  kˆ
Vector Multiplication
Cross Product (magnitude)



The cross product is a way to multiply 2
vectors and get a third vector as an
answer.
The cross product is denoted by:
 
 
A  B  C  A B sin 
The magnitude of the cross product is
the product of the magnitude of B and
the component of A perpendicular to B.
Vector multiplication
Cross product (direction)
Vector Multiplication
Cross product


The vector C represents the solution to the
cross product of A and B.
To find the components of C, use the
following
C x  Ay BZ  Az B y
C y  Az Bx  Ax Bz
CZ  Ax B y  Ay Bx
Vector Multiplication
Cross product

This is more easily remembered using a
determinant
iˆ
 
A  B  Ax
Bx
ˆj
Ay
By
kˆ
Az
Bz
Vector Multiplication
Cross Product Example #1

Vector A has a magnitude of 6 units
and is in the direction of the + x-axis.
Vector B has a magnitude of 4 units
and lies in the x-y plane, making an
angle of 30º with the + x-axis. What is
the cross product of these two
vectors?