Mathematical Concepts:Polynomials, Trigonometry and Vectors

AP Physics C

20 Aug 2009

Polynomials review

“zero order” f(x) = mx0

“linear”: f(x) = mx1 +b “quadratic”: f(x) = mx2 + nx1 + b And so on…. Inverse functions

Inverse

Inverse square

x

axf

2x

axf

Polynomial graphs

Linear

Quadratic

Inverse

InverseSquare

Right triangle trig

Trigonometry is merely definitions and relationships. Starts with the right triangle.

b

ac

bc

a

tan

cos

sin

a

b

c

Special Right Triangles

30-60-90 triangles 45-45-90 triangles 37-53-90 triangles (3-4-5 triangles)

Trigonometric functions & identities

x

x

x

tan

cos

sin

xx

xx

xx

tan

1cot

cos

1sec

sin

1csc

yxxy

yxxy

yxxy

1

1

1

cottan

coscos

sinsin

Trig functionsReciprocal trig

functionsReciprocal trig

functions

Trig identities

x

xxcos

sintan xx 22 cossin1

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 x y y z za a a

r x x y y z za a a

r x i y j z ka a a

or

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 additiongraphical method

+ =

+ =

Vector additionnumerical 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 N57.3 m, 36 S of W17.4 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 MultiplicationDot Product

The dot product (or scalar product), is denoted by:

It is the projection of vector A multiplied by the magnitude of vector B.

cosBABA

Vector multiplicationDot product

In terms of components, the dot product can be determined by the following:

zzyyxx BABABABA

Vector multiplicationDot product Example #1

Find the scalar product of the following two vectors. A has a magnitude of 4, B has a magnitude of 5.

53º50º

A

B

Vector MultiplicationDot Product Example #2

Find the angle between the two vectors

kjiB

kjiA

ˆˆ2ˆ4

ˆˆ3ˆ2

Vector MultiplicationCross 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:

The magnitude of the cross product is the product of the magnitude of B and the component of A perpendicular to B.

sinBACBA

Vector multiplicationCross product (direction)

Vector MultiplicationCross product

The vector C represents the solution to the cross product of A and B.

To find the components of C, use the following

xyyxZ

zxxzy

yzZyx

BABAC

BABAC

BABAC

Vector MultiplicationCross product

This is more easily remembered using a determinant

zyx

zyx

BBB

AAA

kji

BA

ˆˆˆ

Vector MultiplicationCross 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?