(1)+Dimensions,+Vectors+&+Error+Analysis

AGE EDITOR(S): Ivana Gonzalez, Kiara Cerda, Louise Fischel-Bock, Rony Krell

=Dimensions = == = = = = = = = = = = = = =Vectors =
 * Dimensions we need to know are: length, mass, and time
 * Some measurements combine dimensions.
 * Velocity = L/T (Length / Time)
 * Acceleration = L/T^2 (Length / Time^2)
 * In order for equations to be valid, each term in a certain equation must have the exact same dimensions.
 * For example: V= 3A (or velocity equals 3*acceleration) is not valid because velocity does not have dimensions identical to acceleration.
 * 2342321.43504(V/T) =392340932840298(A) is valid because each term has identical dimensions. The coefficients or "numbers" before each term do not make a difference in "dimensional validity".
 * [[image:Page_7.jpg width="659" height="951"]]

> > > > > > > > > > > > > > > > > > __[|3]__.
 * A vector is a type of measurement that illustrates a force's magnitude and direction. It is drawn as an arrow (whose length depends on the magnitude of the force) pointed according to where the force is directed. For example, Vector A has a magnitude of 5 Newtons pointed 45 degrees north of east. It is shown like this
 * A vector can have a negative magnitude depending on which direction is perceived as positive. For example, when dropping a ball from above, its magnitude is perceieved as negative. In general, movement towards the origin is perceieved as negative and movement away from the origin is perceived as positive.
 * A vector may have either one or two component vectors (in a two dimensional situation). Each component vector can be created by "shining a flashight" at the vector from a direction (up, down, left, or right) and drawing a straight line on the opposite side to indicate the vector's displacement. By drawing two component vectors one can form a triangle with a vector and calculate magnitudes and angles using trigonometry.
 * In a situation where you must add or subtract vectors, the components of the resultant vector, "Vector R", usually follow the following equation: R(x) = A(x) +/- B(x) +/- C(x) etc. and R(y)= A(y) +/- B(y) +/- C(y) etc. Note that each component's sign must be taken into account. [[image:Page_9.jpg width="498" height="567" align="center"]]
 * Figure __[|2]__ (a) shows that **A + B = B + A**. The sum of the vectors is called the **resultant** and is the diagonal of a parallelogram with sides **A** and **B.** Figure __[|2]__ (b) illustrates the construction for adding four vectors. The resultant vector is the vector that results in the one that completes the polygon.

<span style="color: #57585b; font-family: Helvetica,Verdana,Arial,sans-serif; font-size: 11px; line-height: normal;">Vector AdditionThe easiest way to learn how vector addition works is to look at it graphically. There are two equivalent ways to add vectors graphically: the **tip-to-tail method** and the**parallelogram method**. Both will get you to the same result, but one or the other is more convenient depending on the circumstances.**Tip-to-Tail Method**We can add any two vectors, <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// and <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**//, by placing the tail of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// so that it meets the tip of<span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**//. The sum, <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**//, is the vector from the tail of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// to the tip of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**//. Note that you’ll get the same vector if you place the tip of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// against the tail of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**//. In other words, <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + //**B**// and <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// + //**A**// are equivalent.**Parallelogram Method**To add <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// and <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// using the parallelogram method, place the tail of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// so that it meets the tail of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**//. Take these two vectors to be the first two adjacent sides of a parallelogram, and draw in the remaining two sides. The vector sum, <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + //**B**//, extends from the tails of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// and <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// across the diagonal to the opposite corner of the parallelogram. If the vectors are perpendicular and unequal in magnitude, the parallelogram will be a rectangle. If the vectors are perpendicular and equal in magnitude, the parallelogram will be a square. **Adding Vector Magnitudes**Of course, knowing what the sum of two vectors looks like is often not enough. Sometimes you’ll need to know the magnitude of the resultant vector. This, of course, depends not only on the magnitude of the two vectors you’re adding, but also on the angle between the two vectors.**Adding Perpendicular Vectors**Suppose vector <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// has a magnitude of 8, and vector <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// is perpendicular to <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// with a magnitude of 6. What is the magnitude of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// ? Since vectors <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// and <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// are perpendicular, the triangle formed by <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**//, <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// , and <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// is a right triangle. We can use the Pythagorean Theorem to calculate the magnitude of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**//, which is  **Adding Parallel Vectors**If the vectors you want to add are in the same direction, they can be added using simple arithmetic. For example, if you get in your car and drive <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">eight miles east, stop for a break, and then drive <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">six miles east, you will be <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">8 + 6 = 14 miles east of your origin. If you drive eight miles east and then six miles west, you will end up <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">8 – 6 = 2 miles east of your origin. **Adding Vectors at Other Angles**When <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// and <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// are neither perpendicular nor parallel, it is more difficult to calculate the magnitude of <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// because we can no longer use the Pythagorean Theorem. It is possible to calculate this sum using trigonometry. **EXAMPLE**

First, add the two parallel vectors, <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// and <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**//. Because they are parallel, this is a simple matter of straightforward addition: 9 + 3 = 12. So the vector <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// has a magnitude of 12 and points due north. Next, add <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// to <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**C**//. These two vectors are perpendicular, so apply the Pythagorean Theorem: <span style="color: #333333; font-family: georgia,times,fantasy; font-size: 14px; line-height: normal;">The sum of the three vectors has a magnitude of 13. > > Read more: __[]__
 * |||| Vector <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// has a magnitude of 9 and points due north, vector <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// has a magnitude of 3 and points due north, and vector <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**C**// has a magnitude of 5 and points due west. What is the magnitude of the resultant vector, <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**C**// ? ||  ||
 * |||| Vector <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// has a magnitude of 9 and points due north, vector <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// has a magnitude of 3 and points due north, and vector <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**C**// has a magnitude of 5 and points due west. What is the magnitude of the resultant vector, <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**A**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**B**// + <span class="question_inline" style="font-family: 'Times New Roman',Times,serif; font-size: 12px; letter-spacing: 0.05em;">//**C**// ? ||  ||

> <span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 14px; line-height: normal;">This picture illustrates vector subtraction. To subtract A and B you just add positive A and -B. Therefore A - B= D. >

> > > > This picture illustrates the tail to head method of adding vectors. Vector C is the sum of Vectors A and B. >

=<span style="font-family: Arial,Helvetica,sans-serif; font-weight: 800;"> <span style="font-family: arial,helvetica,sans-serif; font-size: 13px; font-weight: normal; line-height: 19px;"> = If you really don't understand adding vectors with components: http://www.youtube.com/watch?v=u5iZbY_-kvA http://www.youtube.com/watch?v=JM4pgrEZDFI&feature=related
 * =<span style="font-family: arial,helvetica,sans-serif; font-size: 13px; font-weight: normal; line-height: 19px;">[[image:Page_2.jpg width="513" height="681" align="center"]] [[image:Page_1.jpg width="658" height="728" align="center"]] =

media type="youtube" key="xJBGfPfE4fQ" height="344" width="425"
Video defines vectors and their purpose. Video also breaks down addition and subtraction of vectors and how one does that.Disregard after 2:10 = = =<span style="font-family: Arial,Helvetica,sans-serif; font-weight: 800;">Error Calculation = = =

> % error = 100*|actual value - experimental value|/actual value >

=<span style="font-family: Arial,Helvetica,sans-serif;">Websites =


 * ======http://www.physicsclassroom. com/Class/1DKin/U1L1b.cfm======
 * ======Provides a definition of scalars and vectors and some practice problems to check your understanding.======


 * http://www.colorado.edu/ physics/phys2010/phys2010_ sm99/NOTES/Lecture3.html
 * ======Provides a step by step example of adding and subtracting vectors graphically.======


 * ======http://www.mathwarehouse.com/ vectors/======
 * ======Provides practice problems with the answers explained for extra practice.======
 * ======http://www.glenbrook.k12.il. us/gbssci/Phys/Class/vectors/ u3l1a.html======
 * ======Provides another explanation of vectors and includes animated tutorials for further explanations.======

=
Pictures     ======

Video media type="custom" key="4413335" width="160" height="160" http://www.5min.com/Video/Base-Jumping-in-Terminal-Velocity-97275667

Besides the very cool aerial shots, I chose this video because the people reach terminal velocity (around 120 mph) while flying off ridiculously huge mountains

**Natland Note (9/21/09):** Good stuff guys. The video isn't as related, but I like it anyway. is it possible to include some pictures showing vector addition? e.g. if you follow this link: [], there is a picture or two showing vector addition. Or you can type "vector addition" into google images and see what you find.

<span style="font-family: Arial,Helvetica,sans-serif;">SOURCES**

Vector addition examples are taken from []http://www.sparknotes.com/testprep/books/sat2/physics/chapter4section2.rhtml