!Quadratic Formula\n\nFor an equation of the form:\n\s[ax^2+bx+c=0\s]\n\s[x=\sfrac{-b}{2a}\spm\sfrac{\ssqrt{b^2-4ac}}{2a}\s]\n\n!Binomial Formula\n\n\s[(x+y)^n=\ssum_{k=0}^n \sleft(\sbegin{array}{c} n \s\s k \send{array}\sright)x^ky^{n-k} \s]\n\s[\sleft(\sbegin{array}{c} n \s\s k \send{array}\sright)=\sfrac{n!}{(n-k)!k!} \s]\n\nReference: [[Mathworld: Binomal Identity|http://mathworld.wolfram.com/BinomialIdentity.html]]

!Derivative Rules\n//Please note that these all assume that the function is differentiable at all points, and $x\sin\smathbf{C}$//\n\n|>|! Product Rule |\n|>| \s[\sfrac{d }{dx}f(x)g(x)=\sfrac{df(x)}{dx}g(x)+f(x)\sfrac{dg(x) }{dx} \s] |\n|>|! Quotient Rule |\n|>| \s[\sfrac{d }{dx}\sfrac{f(x)}{g(x)}=\sfrac{\sfrac{df(x)}{dx}g(x)-f(x)\sfrac{dg(x) }{dx}}{g(x)^2} \s] |\n|>|! Chain Rule |\n| \s[\sfrac{d }{dx}f(g(x))=\sfrac{df(g(x))}{dx}\sfrac{dg(x)}{dx} \s] | or \s[\sfrac{dy(u)}{dx}=\sfrac{dy}{du}\sfrac{du}{dx} \s] |\n|>|! Full Derivative |\n|>| \s[\sfrac{d }{dx}f(x_0,x_1,\scdots,x_n)=\sfrac{df}{dx_0}\sfrac{\spartial x_0}{\spartial x}+\sfrac{df}{dx_1}\sfrac{\spartial x_1}{\spartial x}+\scdots+\sfrac{df}{dx_n}\sfrac{\spartial x_n}{dx}+\sfrac{df}{dx} \s] |\n\n!Table of Derivatives\n\n|>|>|! Power of x |\n|>|>| \s[\sfrac{d }{dx}x^n=nx^{n-1}\s] |\n|>|>|! Exponential/Logarithmic |\n| \s[\sfrac{d }{dx}e^x=e^x\s] | \s[\sfrac{d }{dx}a^x=\sln(a)a^x\s] | \s[\sfrac{d }{dx}\sln(x)=\sfrac{1}{x}\s] |\n|>|>|! Trigonometric |\n| \s[\sfrac{d }{dx}\ssin(x)=\scos(x)\s] | \s[\sfrac{d }{dx}\scos(x)=-\ssin(x)\s] | \s[\sfrac{d }{dx}\stan(x)=\ssec^2(x)\s] |\n| \s[\sfrac{d }{dx}\scsc(x)=-\scsc(x)\scot(x)\s] | \s[\sfrac{d }{dx}\ssec(x)=\ssec(x)\stan(x)\s] | \s[\sfrac{d }{dx}\scot(x)=-\scsc^2(x)\s] |\n|>|>|! Inverse Trigonometric |\n| \s[ \sfrac{d }{dx}\sarcsin(x)=\sfrac{1}{\ssqrt{1-x^2}} \s] | \s[ \sfrac{d }{dx}\sarccos(x)=-\sfrac{1}{\ssqrt{1-x^2}}\s] | \s[ \sfrac{d }{dx}\sarctan(x)=\sfrac{1}{1+x^2} \s]|\n| \s[ \sfrac{d }{dx}arccsc(x)=-\sfrac{1}{|x|\ssqrt{1-x^2}} \s] | \s[ \sfrac{d }{dx}arcsec(x)=\sfrac{1}{|x|\ssqrt{1-x^2}}\s] | \s[ \sfrac{d }{dx}arccot(x)=-\sfrac{1}{1+x^2} \s]|\n|>|>|! Hyperbolic |\n| \s[\sfrac{d }{dx}\ssinh(x)=\scosh(x)\s] | \s[\sfrac{d }{dx}\scosh(x)=\ssinh(x)\s] | \s[\sfrac{d }{dx}\stanh(x)=sech^2(x)\s] |\n| \s[\sfrac{d }{dx}csch(x)=-csch(x)coth(x)\s] | \s[\sfrac{d }{dx}sech(x)=-sech(x)\stanh(x)\s] | \s[\sfrac{d }{dx}coth(x)=-csch^2(x)\s] |

!Integral Rules\n\n|! Integrate By Parts (inverse product rule) |\n| \s[\sint_a^budv=uv|_a^b-\sint_a^bvdu \s] |\n|! Fundamental Theorem of Calculus ($\sint f(x)dx = F(x)$) |\n| \s[\sfrac{d }{dx}\sint_a^bf(x)dx=F(b)-F(a)\s] |\n\n!Table of Integrals (Indefinite)\n\n|>|>|! Powers of x |\n|>|>| \s[\sint x^ndx=\sfrac{x^{n+1}}{n+1}\s] $n\sneq1$ |\n|>|>|! Logarithmic/Exponential |\n| \s[\sint e^xdx=e^x \s] | \s[\sint a^xdx=\sfrac{a^x}{\sln a}\s] | \s[\sint\sln(x)dx=x\sln(x)-x\s] |\n|>|>| \s[\sint\sfrac{df(x)}{f(x)}=\sln{f(x)}\s] |

There turns out to be another way to combine our vectors. Remember back when I said that you dropped the cross terms if you were going to use the FOIL method? Well, the cross product, algebraically is where those terms went.\n\nThis one's a little weird to get a hold of algebraically, so an example is definitely in order. Take $\smathbf{A}=a\shat{x}+b\shat{y}$ and $\smathbf{B}=c\shat{x}+d\shat{y}$.\n\n\s[\smathbf{A}\stimes\smathbf{B}=(ad-bc)\shat{z}\s]\n\nOkay. ''Now'' I hear you groaning. Where in the hell did that $\shat{z}$ come from?! Well, it turns out that when you cross two vectors, whatever the answer you get from taking the cross terms you have to append them to a vector that is //perpendicular// to both of them. Yeah, that's a little out there.\n\nAlso take note that you can arrive at the cross terms by using a determinant (which you may or may not remember from Algebra 2 or so). When dealing with vectors with more than two components, a determinant is the prefered way to deal with cross products. That would look something like this:\n\n\s[\smathbf{A}\stimes\smathbf{B}=\sleft(\sbegin{array}{ccc} \shat{x} & \shat{y} & \shat{x} \s\s a & b & c \s\s d & e & f \send{array}\sright)\s]\n\nThe components of vector $\smathbf{A}$ are along the second row and $\smathbf{B}$ are along the third.\n\nSo, okay, what does this all ''mean'' then? Well, if the DotProduct is a measure of how parallel two vectors are, I bet you can guess what the cross product is...\n\nYep. It's a measure of how perpendicular two vectors are. Okay, next step. If we used $\scos$ for our dot product... then what should we use for the cross product which does the exact opposite for parallel and perpendicular vectors? Alright, $\ssin$ it is.\n\nGeometrically:\n\n\s[\smathbf{A}\stimes\smathbf{B}=|\smathbf{A}||\smathbf{B}|\ssin(\stheta)\s]\n\nKeep in mind, that's only the magnitude. To get the direction, you'll have to find a vector direction that perpendicular to both. A little thought will tell you that there are two choices for this vector (take the xy plane, a vector in the $\shat{z}$ direction or in its negative will both do the job). Which one is the right one? You just answered your own question. It's called the RightHandRule, and it essentially tells you that if you point you fingers down the first vector and curl your fingertips towards the second vector (the order is important!) your thumb will point in the correct direction. This is all a consequence of our right handed coordinate systems (yeah, sucks to be a lefty, with all this prejudice and all) which you can learn more about from the RghtHandRule section.\n\nOkay. Summary time.\n\n|>|! Cross Product |\n| Geometrically | A measure of how perpendicular two vectors are, with that information appended to a vector which is perpendicular to both, or \s[\smathbf{A}\stimes\smathbf{B}=|\smathbf{A}||\smathbf{B}|\ssin(\stheta)\s] |\n| Algebraically | \s[(a\shat{x}+b\shat{y})\stimes(c\shat{x}+d\shat{y})=(ad-bc)\shat{z}\s] |

StartHere

Well, we know this much:\n\n\s[\smathbf{C}\scdot\smathbf{C}=|\smathbf{C}|^2\s] by our definition of magnitude\n\n...and since we know that $\smathbf{C}$ is just $\smathbf{A}-\smathbf{B}$. We can also say:\n\n\s[\smathbf{C}\scdot\smathbf{C}=(\smathbf{A}-\smathbf{B})^2\s]\n\nNow, this ''is'' the FOIL rule, so we can expand that out to be:\n\n\s[\smathbf{A}^2+\smathbf{B}^2-2\smathbf{A}\scdot\smathbf{B}\s]\n\nNotice that $\smathbf{A}\scdot\smathbf{B}=\smathbf{B}\scdot\smathbf{A}$. It inherited that from the way we mulitply numbers (associtivity).\n\nSo, we have the magnitude squared of $\smathbf{C}$ on the left, and the magnitude square of $\smathbf{A}$ and $\smathbf{B}$ on the right, as well as the dot product on the end. If we expand out the dot product into the geometric interpretation, we're done:\n\n\s[c^2=a^2+b^2-2ab\scos(\stheta)\s]\n\nYou simply can't escape this formula in physics, so get used to it.

Okay. So we've spoken a lot of this magnitude and direction thing. Geometrically, it's easy to measure, get a ruler out and measure the length. Then pick a reference point and measure how far off from the reference the vector points.\n\nNow. Things aren't so easy with our algebraic forms. So, before we jump into this, let's take another look at that multiplication thing. Suppose we considered our vector to be like a binomal (you remember, like $(a+b)(c+d)$)? So, we can use the FOIL rule here to get something that looks like this:\n\n\s[\smathbf{A}\scdot\smathbf{B}=(3+4)(3+4)=7\sdot7=49\s]\n\nWell... you're close. It turns out the that cross terms (the terms that arise from the middle two multiplications) don't count. They would if we were being very general (that's called an outer product) but in our case it simplifies down because of the more advanced properties of UnitVectors (check the link out if you want a further explanation, but's not necessary for this level).\n\n\s[\smathbf{A}\scdot\smathbf{B}=(3)(4)+(4)(3)=14\s]\n\nOkay! Now we're getting somewhere. Note that the dot product gives you a number... a scalar. This is important in more advanced treatements of vectors, so keep it in mind if you've got an eye to continue on to higher levels.\n\nThat makes sense algebraically, but what does it have to do with the geometrical interpretation? Well, the dot product of the two vectors gives you a measure of how parallel they are. So, we need something that will give us 0 if the two vectors are perpendicular (some angle between them is 90) and 1 if the two vectors are parallel (some angle between them is 0). Well, we can use the $\scos$ function for this purpose, and in addition to giving us the two values, it will also tell us all the values for angles in between! It's a bargain, seriously.\n\n\s[\smathbf{A}\scdot\smathbf{B}=|A||B|\scos(\stheta)\s]\n\nWhere the | | will be explained shortly and $\stheta$ is the angle between the two vectors if you put them tail to tail.\n\nMagnitude, magnitude... I promised you a magnitude and here it is. We can take the dot product of a vector with itself and then take the square root (check out the pythagorean theorem for why we might have to take a root in there).\n\n\s[|\smathbf{A}|=\ssqrt{\smathbf{A}\scdot\smathbf{A}}=\ssqrt{3^2+4^2}=5\s]\n\nLet's summarize.\n\n|>|! Dot Product |\n| Geometrically | Number indicating how parallel two vectors are. |\n| Algebraically | Multiplication of two vectors |\n\nExercise: The dot products of any two vectors are the same number regardless of the interpretation used to obtain them. Use this fact to derive the Law of Cosines: $c^2=a^2+b^2-2ab\scos(\stheta)$ where a, b, and c are the magnitudes of vectors $\smathbf{A}$, $\smathbf{B}$, and $\smathbf{A}-\smathbf{B}=\smathbf{C}$ and $\stheta$ is the angle between $\smathbf{A}$ and $\smathbf{B}$.\n[[Answer|DotProdAns1]]

// AJAX code adapted from http://timmorgan.org/mini\n// This is already loaded by ziddlywiki...\nif(typeof(window["ajax"]) == "undefined") {\n ajax = {\n x: function(){try{return new ActiveXObject('Msxml2.XMLHTTP')}catch(e){try{return new ActiveXObject('Microsoft.XMLHTTP')}catch(e){return new XMLHttpRequest()}}},\n gets: function(url){var x=ajax.x();x.open('GET',url,false);x.send(null);return x.responseText}\n }\n}\n\n// Load jsMath\njsMath = {\n Setup: {inited: 1}, // don't run jsMath.Setup.Body() yet\n Autoload: {root: new String(document.location).replace(/[^/]*$/,'jsMath/')} // URL to jsMath directory, change if necessary\n};\nvar jsMathstr;\ntry {\n jsMathstr = ajax.gets(jsMath.Autoload.root+"jsMath.js");\n} catch(e) {\n alert("jsMath was not found: you must place the 'jsMath' directory in the same place as this file. "\n +"The error was:\sn"+e.name+": "+e.message);\n throw(e); // abort eval\n}\ntry {\n window.eval(jsMathstr);\n} catch(e) {\n alert("jsMath failed to load. The error was:\sn"+e.name + ": " + e.message + " on line " + e.lineNumber);\n}\njsMath.Setup.inited=0; // allow jsMath.Setup.Body() to run again\n\n// Define wikifers for latex\nconfig.formatterHelpers.mathFormatHelper = function(w) {\n var e = document.createElement(this.element);\n e.className = this.className;\n var endRegExp = new RegExp(this.terminator, "mg");\n endRegExp.lastIndex = w.matchStart+w.matchLength;\n var matched = endRegExp.exec(w.source);\n if(matched) {\n var txt = w.source.substr(w.matchStart+w.matchLength, \n matched.index-w.matchStart-w.matchLength);\n if(this.keepdelim) {\n txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);\n }\n e.appendChild(document.createTextNode(txt));\n w.output.appendChild(e);\n w.nextMatch = endRegExp.lastIndex;\n }\n}\n\nconfig.formatters.push({\n name: "displayMath1",\n match: "\s\s\s$\s\s\s$",\n terminator: "\s\s\s$\s\s\s$\s\sn?",\n element: "div",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nconfig.formatters.push({\n name: "inlineMath1",\n match: "\s\s\s$", \n terminator: "\s\s\s$",\n element: "span",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nvar backslashformatters = new Array(0);\n\nbackslashformatters.push({\n name: "inlineMath2",\n match: "\s\s\s\s\s\s\s(",\n terminator: "\s\s\s\s\s\s\s)",\n element: "span",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nbackslashformatters.push({\n name: "displayMath2",\n match: "\s\s\s\s\s\s\s[",\n terminator: "\s\s\s\s\s\s\s]\s\sn?",\n element: "div",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nbackslashformatters.push({\n name: "displayMath3",\n match: "\s\s\s\sbegin\s\s{equation\s\s}",\n terminator: "\s\s\s\send\s\s{equation\s\s}\s\sn?",\n element: "div",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\n// These can be nested. e.g. \sbegin{equation} \sbegin{array}{ccc} \sbegin{array}{ccc} ...\nbackslashformatters.push({\n name: "displayMath4",\n match: "\s\s\s\sbegin\s\s{eqnarray\s\s}",\n terminator: "\s\s\s\send\s\s{eqnarray\s\s}\s\sn?",\n element: "div",\n className: "math",\n keepdelim: true,\n handler: config.formatterHelpers.mathFormatHelper\n});\n\n// The escape must come between backslash formatters and regular ones.\n// So any latex-like \scommands must be added to the beginning of\n// backslashformatters here.\nbackslashformatters.push({\n name: "escape",\n match: "\s\s\s\s.",\n handler: function(w) {\n w.output.appendChild(document.createTextNode(w.source.substr(w.matchStart+1,1)));\n w.nextMatch = w.matchStart+2;\n }\n});\n\nconfig.formatters=backslashformatters.concat(config.formatters);\n\nwindow.wikify = function(source,output,highlightRegExp,tiddler)\n{\n if(source && source != "") {\n var wikifier = new\n Wikifier(source,formatter,highlightRegExp,tiddler);\n wikifier.subWikify(output,null);\n jsMath.ProcessBeforeShowing();\n }\n}\n

StartHere\n*Chapter 1 topics\n**[[Vectors|Vectors]]\nReferenceSheets

All in units of SI unless otherwise noted.\n\n|>|! Gravity |\n|$G=6.6742\stimes10^{-11} \sfrac{m^3}{kg s^2}$ | Gravitational constant |\n|$g=\sfrac{GM_e}{R_e^2}=9.80665 \sfrac{kg m}{s^2}$ | Gravitational acceleration at the surface of the earth (sea level) |\n\n|>|! Electrodynamics |\n| $m_e=9.1093826\stimes10^{-31} kg$ | Electron mass |\n| $m_p=1.67262171\stimes10^{-27} kg$ | Proton mass |\n| $1 eV=1.60217653\stimes10^{-19} J$ | Electron volt |\n| $\sepsilon_0=8.854187\stimes10^{-12} \sfrac{F}{m}$ | Permittivity of free space |\n| $\smu_0=4\spi\stimes10^-7 \sfrac{N}{A^2}$ | Permissivity of free space |\n| $e=1.60216753\stimes10^{-19} C$ | Electron charge |\n| $c=2.99792458\stimes10^{8} \sfrac{m}{s}$ | Speed of light in vacuum $\sfrac{1}{\ssqrt{\smu_0\sepsilon_0}}$ |\n| $\salpha=7.297352568\stimes10^{-3}=\sfrac{1}{137.0359991} unitless$ | Fine structure constant $\sfrac{e^2}{4\spi\sepsilon_0\shbar c}$ |\n\n|>|! Quantum mechanics |\n| $\shbar=1.05457168\stimes10^{-34} J s$ | Planck constant $\sfrac{h}{2\spi}$ |\n| $R_\sinfty=1.0973731568\stimes10^7 \sfrac{1}{m}$ | Rydberg constant |\n\n|>|! Thermodynamics and Statistical Mechanics |\n| $k_B=1.3806505\stimes10^{23} \sfrac{J}{K}$ | Boltzmann constant |\n| $N_A=6.0221415\stimes10^{23} \sfrac{1}{mol}$ | Avagadro's number |\n\nReference: [[Fundamental Physical Constants from NIST|http://physics.nist.gov/cuu/Constants/index.html]]

Reference Sheets for\nAlgebra: AlgEquationSheet\nTrigonometry: TrigEquationSheet\nCalculus: CalcDerivEquationSheet\n\nConstants: PhysicalConstants

Class Journal for PHYS at the [[University of Florida|http://www.ufl.edu]]

~PhysWiki

Welcome!\n\nThis will be a kind of online notebook for my physics classes. It will contain both introductory, intermediate, and advanced information (hopefully at some point) but for the benefit of my students I'll start at the introductory level.\n\nThe best way to use this is to either search for a topic on the right side text box. Or to select a topic from the left menu bar.\n\nYou can expand as many topics as you like and you can print it it as a reference sheet if you need one.\n\nYou can even save this webpage (File->Save As...) and have this entire thing as a kind of mininotebook. It will work offline just as well as online.

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This is a test\n\s[-\sfrac{\shbar^2}{2m}\snabla^2\sPsi+V(x)\sPsi=i\shbar\sfrac{\spartial\sPsi}{\spartial t}\s]

!Plane Trigonometry\n\nFor a right triangle with sides a, b, and r.\n\n|>|>|! Pythagorean Theorem |\n|>|>| \s[a^2+b^2=r^2\s] |\n|>|>| \s[\stheta=\sarccos\sleft(\sfrac{x}{\ssqrt{x^2+y^2}}\sright)=\sarcsin\sleft(\sfrac{y}{\ssqrt{x^2+y^2}}\sright)=\sarctan\sleft(\sfrac{y}{x}\sright)\s] |\n|>|>|! Basic formulas |\n| \s[cos(\stheta)=\sfrac{x}{\ssqrt{x^2+y^2}}=\sfrac{x}{r}\s] | \s[sin(\stheta)=\sfrac{y}{\ssqrt{x^2+y^2}}=\sfrac{y}{r}\s] | \s[tan(\stheta)=\sfrac{y}{x}\s] |\n| \s[sec(\stheta)=\sfrac{\ssqrt{x^2+y^2}}{x}=\sfrac{r}{x}\s] | \s[csc(\stheta)=\sfrac{\ssqrt{x^2+y^2}}{y}=\sfrac{r}{y}\s] | \s[cot(\stheta)=\sfrac{x}{y}\s] |\n|>|>|! Square Relations (note that you can get the second two from the first by dividing by $\ssin$ or $\scos$) |\n| \s[\scos^2(\stheta)+\ssin^2(\stheta)=1\s] | \s[1+\stan^2(\stheta)=\ssec^2(\stheta)\s] | \s[\scot^2(\stheta)+1=\scsc^2(theta)\s] |\n|>|>|! Addition Formulas |\n|>|>| \s[\scos(\stheta+\sphi)=\scos(\stheta)\scos(\sphi)-sin(\stheta)\ssin(\sphi)\s] |\n|>|>| \s[\ssin(\stheta+\sphi)=\scos(\stheta)\ssin(\sphi)+\scos(\sphi)\ssin(\stheta)\s] |\n|>|>|! Special Cases of Addition Formula (set $\sphi=\stheta$) |\n|>|>| \s[\ssin(2\stheta)=2\scos(\stheta)\ssin(\stheta)\s] |\n|>|>| \s[\scos(2\stheta)=\scos^2(\stheta)-\ssin^2(\stheta)\s] |\n\nThis should be all you need. From here on out, it gets messy.\n\n|>|! Euler's Formula |\n|>| \s[\sexp^{\simath\stheta}=\scos(\stheta)+\simath\ssin(\stheta)\s] |\n|>|! Euler $\ssin$ and $\scos$ formulas |\n| \s[\ssin(\stheta)=\sfrac{e^{\simath\stheta}-e^{-\simath\stheta}}{2}\s] | \s[\scos(\stheta)=\sfrac{e^{\simath\stheta}+e^{-\simath\stheta}}{2}\s] |\n\n|! Multiple Angle Formula |\n| \s[\ssin(nx)=\ssum_{k=0}^n \sleft(\sbegin{array}{c}n \s\s k \send{array}\sright) \scos^k(x)\ssin^{n-k}(x)\ssin \sleft[ \sfrac{1}{2}(n-k)\spi \sright] \s] |\n| \s[\scos(nx)=\ssum_{k=0}^n\sleft(\sbegin{array}{c}n \s\s k \send{array}\sright) \scos^k(x)\ssin^{n-k}(x)\scos \sleft[ \sfrac{1}{2}(n-k)\spi \sright] \s] |\n\n|! Law of Cosines |! Law of Sines |\n| \s[c^2=a^2+b^2-2ab\scos(\stheta)\s] | \s[\sfrac{\ssin(A)}{A}=\sfrac{\ssin(B)}{B}=\sfrac{\ssin(C)}{C}\s] |\n\n!Hyperbolic Trigonometry\n\n|>|>|! Basic formulas |\n| \s[\scosh(z)=\scos(\simath z)\s] | \s[\ssinh(z)=\simath\ssin(\simath z)\s] | \s[\stanh(z)=\sfrac{\ssinh(z)}{\scosh(z)}\s] |\n|>|>|! Square Relations (note that you can get the second two from the first by dividing by $\ssinh$ or $\scosh$) |\n| \s[\scosh^2(z)-\ssinh^2(z)=1\s] | \s[1-\stanh^2(z)=sech^2(z)\s] | \s[\scoth^2(z)-1=csch^2(z)\s] |\n\nReferences: [[Mathworld|http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html]]

Displacement = $\smathbf{0}$ (that's the zero vector not 0, there is a difference) Because you ended up in the same place you started\n\nDistance = 2 miles to, 2 miles back = 4 miles total

Remember what we said about being able to represent a vector any way we wanted? That means it doesn't have to be tied down in space. We can drag it anywhere we want as long as we keep the magnitude and direction the same.

!Interpretation of Vectors\n\nA vector is a very strange object when you get right down to it. It has the advantage of being free of interpretation. This seems a little counter to the title of this section, but it actually makes things easier on the student and the physicist. It essentially means that we can define a way to express a vector, and as long as we follow the rules of how that definition works, every vector can be expressed under that definition. Now, we could go ahead and define displacement in some strange frame of reference (like relative to the position of Alpha Centauri or to five miles to the right of the second moon of Saturn) and it will be [[correct]] but not necessarily useful (unless we're travelling to Alpha Centauri). So, here comes the punchline: We'll define a vector under any auspices that make the problem at hand convenient to do.\n\nIn practical physics, this admits a few interpretations, so let's look at the big two: what I can Algebraic and Geometric.\n\nLet's take a bike ride, shall we? Let's say that you ride your bike from your house (or cardboard box, for all the underpaid physicists out there) to the park. It's 2 miles to the park and it's in a direction that we'll call northwest.\n\nI want you to think about your displacement in these cases.\n\n\n!!Geometric Interpretation\n\nThis one you may be familiar with already. You can express the vector by drawing an arrow. Its length indicates its magnitude and its direction is indicated by where its pointing. A vector that has twice the magnitude of another will be twice as long and a vector that's perpendicular to another will point 90 degrees away from it (note that there are two ways you can do that in a plane as well).\n\nIn our case, the displacement vector would require a map (yes, I said we could do it any way we want, but I'm now choosing a reference, and that requires landmarks, so to speak). We'd draw an arrow starting at your house/box and ending at the park.\n\nSimple, no?\n\n!!Algebraic Interpretation\n\nThis one isn't the one that's well known, but it's used almost exclusively to the geometric interpretation among mathematicians and physicists. It's the one that you'll use as well. Essentially, we can define a vector according to its //components//. \n\nIn this case, we don't need a map (but we still need a reference. I'll get to the connection between the two interpretations in due time, but for now I just want you to see how a vector looks written down on paper.\n\nIf we define the north to be along the positive y-axis, and the east to be along the x-axis and center this crazy looking coordinate system at our house/box, we can define that vector to be:\n\n\s[\sfrac{\ssqrt{2}}{2}\shat{x}-\sfrac{\ssqrt{2}}{2}\shat{y}\sqquad mi\s]\n\nWhoa! Where did all of these funny symbols and square roots come from!? Well, take a closer look. The square roots may be weird looking but they're still numbers. The funny looking x's and y's are what we call [[unit vectors|Unit Vectors]].\n\nThat deserves a convention.\n|! Convention | $\shat{x}$,$\shat{y}$,$\shat{z}$ stand for unit vectors in the xyz plane |\n|~| Other possibles are $\shat{i}$,$\shat{j}$,$\shat{k}$. They're used with similar frequency. I don't like them because of their resemblance to the unit imaginary $\simath$ which can also be represented as a vector. |\n\nThey work just like normal vectors and they're a little indication //how far// in that direction the vector goes. So you'd go $\sfrac{\ssqrt{2}}{2}$ miles north and then $\sfrac{\ssqrt{2}}{2}$ miles in the negative east direction (err.... west), and voila... you're at the park, ready to play frisbee.\n\nNow, a little exercise and review about vectors and scalars. Suppose you're tired of tossing the B around and you want to go home. Once you've arrived back at your house/box, what's your distance? Displacement?\nRemember: displacement is the magnitude of the line between where you started and ended up, distance is the total mileage you incur getting between point A and point B.\n[[Answer|VecIntAns1]]\n\nSo... we know what it is... now what do we do with it? VectorOperations.

!Vector Operations\n\nSo. You know how to take numbers add, subtract, multiply, divide, exponentiate, pontificate, expatriate... err... I'm getting ahead of myself. Now we ask ourselves, can we do the same kinds of things with vectors.\n\nThe answer is... to some extent, yes. We talked about the components of a vector before. Since the components of a vector are just numbers, if we can add and subtract numbers, we should be able to add and subtract components as well, right?\n\nSure. As long as we keep our components straight. Let's suppose we have two vectors $\smathbf{A}=3\shat{x}+4\shat{y}$ and $\smathbf{B}=4\shat{x}+3\shat{y}$ (I know, creativity is at a premium here, sorry).\n\n|>|! Addition of Vectors |\n| Geometrically | Take the tail of $\smathbf{B}$ place it at the tip of $\smathbf{A}$ and draw a new vector (call it $\smathbf{C}$ if you'd like it doesn't really matter) from the tail of $\smathbf{A}$ to the tip of $\smathbf{B}$. There's your new vector. |\n| Algebraically | Add component by component: $\smathbf{C}=(3+4)\shat{x}+(4+3)\shat{y}=7\shat{x}+7\shat{y}$ |\n\nOkay. Pop quiz. Why can we take the vector $\smathbf{B}$ and move it to where $\smathbf{A}$ is to add it?\n[[Answer|VecOpAns1]]\n\nNow, you might as what happens when you do this:\n\n\s[\salpha+\smathbf{A}=?\s]\n\n...for some scalar number $\salpha$? Well... make up what you'd like and call it something new, but you'll never see it in physics or math. It's not a defined operation and it can't really happen. If it does, you know you probably went wrong somewhere.\n\nOkay, what about multiplication? Well, suppose we have our vector $\smathbf{A}$ and our scalar $\salpha$. Okay, we can't add them, but can we multiply them? Sure. Same way we did the addition, component by component.\n\n|>|! Scalar Multiplication |\n| Geometrically | Take the length of $\smathbf{A}$ and multiply it by $\salpha$. Draw a new vector with said length and same direction. |\n| Algebraic | $\salpha(3\shat{x}+4\shat{y})=3\salpha\shat{x}+4\salpha\shat{y}$ |\n\nOkay, great. We can add vectors, and multiply by a scalar. Can we multiply by a vector? Well, now we get to the tricky part. The short answer is yes, but there are multiple ways we can combine our vectors (and the number grows with the more components you have). So, what do we do? Well, we have two standard ways of combining the two. Those are our next sections:\n\nDotProduct\nCrossProduct\n\nAnd when you've had your fill of products... we'll move on.\n\n

!What is a Vector?\n\nAbstractly, we can talk about a vector as having two properties. It has a magnitude and a direction. Any interpretation, or any way you write down a vector it has to somehow express these two properties. Essentially, we define this difference because vectors are fundamentally different from numbers and they are manipulated differently. Simple numbers are what scientists and mathematicians refer to as "scalars". They are things like temperature, mass, charge, distance and speed. Whereas vectors are objects like velocity, displacement, force, and acceleration.\n\nNow, you may cry foul on these definitions, or at the very least express a little confusion over two of these quantities which appear to be the same. I'll single them out so as to express my point about the difference between a vector and a scalar. \n\nTake speed and velocity. When you get pulled over for a speeding ticket, the police officer doesn't seem to be too concerned with the difference between velocity and speed, but he can probably tell you both. It's somewhat semantical, but physicists define the number that the police officer writes on the ticket as //speed/// and the combination of that number with the direction you are going as //velocity//. If we wanted to be super picky (which we can... and will), we could even go so far as to define speed as the magnitude of the velocity vector.\n\nLet's take a look at how we can look at vectors:\n\nVectorInterpretations