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# vectors - Solomons Study Notes College Physics 1 Mechanics...

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S olomon’s Study Notes College Physics 1 Mechanics & Heat Fall 2011 Solomon Weiskop PhD [ Vectors ] These Notes cover everything about Vectors you will need to know for a full two-semester course in College Physics.

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Study Notes are available to print out by registering at www.solomonlinetutor.com Solomon Weiskop PhD Solomon’s Tutoring © Copyright 2011
1 1. Trigonometry In order for you to be able to work with vectors (and for other reasons, too) you’ll need to know some trigonometry. All the trig you will need for a full year of Physics is based on the following right-angle triangle: Under the column for Vectors I’ve written the basic formulas in a form that will be most useful in our study of vectors.

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2 Eqs. (i) and (ii) were obtained from the basic formulas by cross- multiplication: cross-multiplication Eqs. (iii) and (iv) were obtained by taking inverses. For instance, Eq. (iv) gives you HYP in terms of ADJ and OPP. To get it from Pythagoras’ Theorem you need to take the “square root”: , which is the inverse of “square”: . By doing this, you “undo” the square on since . However, since you took the square root of one side, you must also take the square root of the other side. This yields Eq. (iv) . Likewise, Eq. (iii) gives you θ in terms of ADJ and OPP. To ge t it from TOA you need to take the “inverse tan” (or “arc tan”) , which is the inverse of tan. By doing this, you undo the tan on since . However since you took the of one side, you must also take the of the other side. This yields Eq. (iii) . Why the basic formulas are more convenient to work with in the form of Eqs. (i) (iv) will become clear as we now begin our discussion of vectors...
3 2. Vectors A vector is like an arrow. It is something that has “magnitude” and “direction”. Magnitude corresponds to the length of the arrow (how big it is). Direction corresponds to which way the arrow is pointing. Here’s a vector: I will indicate that something is a vector by putting an arrow on top: . For the magnitude, I’ll leave off the arrow: . For the direction, I’ll use an angle: . [I write and not just plain θ in order to specify that this is the angle for the vector . Note: Here I’ve chosen the angle that the vector makes with the positive x -axis. I could just as well have chosen the angle that the vector makes with the positive y-axis (or even with the negative x-axis or negative y-axis). We’ll discuss such possibilities in more detail soon.

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4 A vector is described by two numbers (e.g. one for magnitude and one for direction) Magnitude of a vector is always a positive number An example of a vector is Velocity. If I’m driving my car and I want to describe my velocity vector to you, I need to give you two numbers: one for the magnitude (e.g. 45 mph) and one for the direction (e.g. I’m heading 32° North of East).
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vectors - Solomons Study Notes College Physics 1 Mechanics...

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