TO
RCHS students in AP Physics, your summer assignment is to thoroughly
cover this website. You must take notes that will b e turned in the
first day of class. Email me for a thorough note taking sheet at my
RCHS email (it counts as extra credit).

Be
sure to do the summer assignment at the end of the webpage! You will
turn that in to me the first day of school as well.



What
is a Vector?


A Vector
is any quantity that requires both magnitude and direction to completely
describe itself. Quantities in physics that only require magnitude are
termed scalars. Scalars
and Vectors

Examples of Vectors


How
to Name VECTORS!


A vector is usually named
in one of two ways.

Using the POLAR
FORM way requires magnitude and then direction as in
< R, q > (where R is magnitude
and q is the angle from the +x axis).

Using the RECTANGULAR
or COMPONENT method requires an x part and y part (2
dimensions) or for 3 dimensions an "i" (x axis), "j"
(y axis), and "k" (z axis) written as <x,y,k> or
2 i  4j + 7 k (an example for 3 dimensions).

Examples
of Polar Vectors
Polar form always states
the magnitude first and then the direction.
In physics, the direction is usually from the positive x axis and
rotates counterclockwise following trig notation.

Examples
of Component Vectors
Two types of component
notation are common.
1. Using an algebraic
form of ordered pair notation gives <Vx,
Vy> as the components of vector V.

How
to Calculate Components from Polar Form!


We shall discuss only
the x and y components of vectors. Discuss "direction
cosines" for vectors in 3 space with your math teacher.

To
find the x component
first, make sure your angle is given as directed from the + x axis!
Then multiply the magnitude of the vector by the cosine of the angle
so that Vector (x) = Vector Magnitude
* cosine q.

To
find the y component
first, make sure your angle is given as directed from the + x axis!
Then multiply the magnitude of the vector by the sine of the angle
so that Vector (y) = Vector Magnitude
* sine q

Quadrant
IV Example for converting angles as directed from the +x axis: mouse
over to see how!

Quadrant II
Example for converting angles :

How
to Add Vectors


One of the most important
uses of the component form is adding two or more vectors. To
add vectors:

Find all x components
and all y components.

Add all x components
together keeping the sign of the sum.

Add all y components
together keeping the sign of the sum.

The vector answer has
both an "x" part and a "y" part generally written
as <vx, vy>
or vx i
+ vy j .

To add graphically using
tail to tip method:

Start one vector where
the preceding vector stops. Here vector a (in blue) is added to
vector b (in green).

The RESULTANT
(vector sum) starts at the beginning of the first vector
and terminates at the end of the last vector.

Add these
two vectors:
a = 4 m at 120 degrees
b = 6 m at 210 degrees
Graphicallymouse
over to see sum!
With Component Method

How
to Subtract Vectors


Keep in mind that subtraction
is just a form of addition. To obtain vector
a vector b, just add
vector a to the negative of vector b.
How to take the negative
of a vector:

Graphically, just reverse the direction.

With components, reverse the signs of all parts.

With polar form, add or subtract 180 degrees (it does not matter)
to or from your angle.

Reversing the direction of the vector makes it negative.

Note that vector a is the reverse of vector a. The vector sum
is the red vector.

Converting
to Polar Form


While it is best to have
your vectors in rectangular form for addition, it is also necessary
to be able to convert from rectangular form to polar form.
This is a two part
process.

First, find the magnitude of the vector using the
Pythagorean Theorem. MAGNITUDE
of Vector V=

Next find the reference
angle using trig relationships.
Tangent
q = (y component) /
x component
Note the
absolute values! This gives your reference angle. Now you must find
the correct quadrant! See to the right!

Correct
Direction?
You should
sketch or graph the vector. Use the POSITIVE x axis! IDK
refers to identifying key!

Quadrant
1 IDK
is positive x and positive y ! Your reference angle
is your angle q.

Quadrant
2
IDK is negative x and positive y! Your angle is 180
q.

Quadrant
3 IDK
is negative x and negative y! Your angle is 180
+ q.

Quadrant
4
IDK is positive x and negative y! Your angle is 360
 q
Did
you observe that all angles are relative to the xaxis?

Look at the force vector (blue arrow). Name the force
vector in component form.The unit is Newtons
Now find the magnitude of the force vector in Newtons
Last of all, find the direction of the vector

Multiplying
Vectors


There
are three types of vector multiplication:

Vector
by Scalar:
changes length of a vector by the scalar factor. Direction changes
only if the scalar is negative. Then the direction of the vector reverses
180 degrees when multiplies by a negative scalar. PLEASE NOTE THIS!

Vector "dot"
Vector (APC only): a product that determines how much vectors point
in the same direction. It is a scalar product! It finds the product
of parts or components that point in a specified direction.

Vector "cross"
Vector (APC only): a product that determines how perpendicular vectors
are to each other. This product produces a vector answer! It finds
the product of parts or components that are perpendicular to each
other regardless of original direction.

Equations:

For APC OnlyEquations:
a X b = a
b sine q with direction from right
hand rule

Your
Summer Homework! On
the left sidebar, click on Practice Problems (4th choice down)!
Print out your score!!!!!
PLEASE Note: Only
answer questions 14 & 6 for APB and # 16 for APC and show your
work as well as your answer!






