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.

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

Examples of Scalars

 

 

How to Name VECTORS!

 

 

 

 

 

 

 

 

A vector is usually named in one of two ways.

  1. 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).

  2. 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.
2. Using i to represent the x axis, j to represent the y axis and k to represent the z axis. This gives a vector that has the form Vx i + Vy j + Vz k

 

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.

  1. 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.

  2. 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:

  1. Find all x components and all y components.
  2. Add all x components together keeping the sign of the sum.
  3. Add all y components together keeping the sign of the sum.
  4. 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

Graphically-mouse 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.

  • The red vector is vector a.
  • The blue vector is -a.

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.
  1. First, find the magnitude of the vector using the Pythagorean Theorem.       MAGNITUDE of Vector V=  
  2. 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 x-axis?
 

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:

  1. 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!
  2. 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.
  3. 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.

More on "Dot" Products

Equations:

 

 

 

 

More on "Cross" Products

For APC Only--Equations:

 

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 1-4 & 6 for APB and # 1-6 for APC and show                                                  your work as well as your answer!
For APC Practice Only: Vectorland

What Should I Know?

APB: # 1-4, #6
APC: #1-6
Vectors, An Introduction
More INTRO to Vectors
Vector Basics