There’s another way of working with number-like things that have multiple dimensions in math, which is very different from the complex number family: vectors. Vectors are much more intuitive to most people than the the complex numbers, which are built using the problematic number i. <\/p>\n
A vector is a simple thing: it’s a number with a direction. A car can be going 20mph north – 20mph north is a vector quantity. A 1 kilogram object experiences a force of 9.8 newtons straight down – 9.8n down is a vector quantity.<\/p>\n
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To be precise about the definition, a vector is a quantity<\/em> qualitatively different parts: a magnitude<\/em> (aka, size) and a direction<\/em>.<\/p>\n When we’re talking about vectors, we also generally add the term scalar<\/em>. A scalar is what we were calling a number before we got to vectors – it’s a quantity without<\/em> a direction. 20mph is a scalar quantity called speed<\/em>; 20mph northwest is a vector quantity called velocity<\/em>.<\/p>\n Vectors are fascinating things, which can be used to describe all sorts of physical phenomena. In physics, you simple can’t get away from them – they’re everywhere. Velocity, acceleration, force, momentum – they’re all vectors. <\/p>\n Normally, we draw vectors as arrows<\/em>, where the length of the arrow corresponds to the magnitude of the vector, and the direction of the arrow is (obviously) the direction of the vector. <\/p>\n We generally represent<\/em> vectors in one of two ways. One of them is by length + angle – for example, for a vector in a plane, we might use 10 @ 60 degrees, where the angle is measured relative to the X axis. The other representation is based on components<\/em>. Take that vector 10@60 degrees, and put its starting point on the origin of a 2d graph. Draw a horizontal line from its tip to the y axis – that’s it’s y component, written Ay<\/sub>; draw a vertical line from its tip to the x axis – that’s it’s X component, written Ax<\/sub>. The result is a pair [5.00,8.66]. In general, an n-dimensional vector can be written as n<\/em> scalars between brackets, so a three dimnesional vector could be written [x,y,z], and so on. If we aren’t using the components, we describe the magnitude of a vector A by |A|.<\/p>\n And of course, you can do algebra with vectors. After all, given a new thing Vector algebra is built mainly on addition and two kinds of multiplication.<\/p>\n Vector addition is really amazingly easy. To compute the sum A+B of two vectors A and B, draw them on a graph, with the tip of A touching the tail of B. Then draw an arrow from the tail of A to the tip of B. That arrow is the sum.<\/p>\n Multiplication is a bit weird. There are two different kinds of multiplication of vectors. One, called dot product<\/em> or scalar product<\/em> multiplies two vectors, and results in a scalar. The other, the cross product<\/em> is non-commutative, and only works in at least three dimensions.<\/p>\n Given two vectors A and B, their dot product A⋅B is easiest to define in terms of Σi=1…n<\/sub> Ai<\/sub>*Bi<\/sub><\/p>\n So if A were a vector [A1<\/sub>, A2<\/sub>, A3<\/sub>], and B were The cross product is more interesting. Given two vectors A and B, their cross product A×B is a vector C where C’s magnitude is the area of the parallelogram In three dimension, there’s a simple formula in terms of components. If A=[Ax<\/sub>, Ay<\/sub>, Az<\/sub>], and B=[Bx<\/sub>, By<\/sub>, Bz<\/sub>], then A×B=C where:<\/p>\n So, for example, if A=[3,4,5] and B=[9,7,6] then A×B= Now, as I mentioned, in vector algebra cross-product isn’t commutative. But it’s not a total loss like quaternions; vector multiplication is anti-commutative: A×B = -(B×A).<\/p>\n Cross product isn’t associative either – A×(B×C)≠(A×B)×C. But we do at least have the very interestingly odd property:<\/p>\n A×(B×C) + B×(C×A) + C×(A×B) = 0<\/p>\n \tWhere 0 is the 0 vector – that is, the vector with length 0. (Originally, one of the parens was mispositioned in the above equation; thanks for commenter “rory” for the catch.) <\/em><\/p>\n Finally – we can multiply a scalar times a vector – multiplying a scalar s by a vector A = [x,y,z] gives [sx,sy,sz].<\/p>\n","protected":false},"excerpt":{"rendered":" There’s another way of working with number-like things that have multiple dimensions in math, which is very different from the complex number family: vectors. Vectors are much more intuitive to most people than the the complex numbers, which are built using the problematic number i. 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\nto play with, what do mathematicians do? We invent algebras for it!<\/p>\n
\ntheir components.\tIf they’re n<\/em> dimensional vectors, then the dot product is:<\/p>\n
\na vector [B1<\/sub>, B2<\/sub>, B3<\/sub>], then A⋅B=A1<\/sub>*B1<\/sub>+A2<\/sub>*B2<\/sub>+A3<\/sub>*B3<\/sub>. For example, let’s look at two vectors: A=[2,4,1,5], and B=[6,2,3,-4]. A⋅B=2*6+4*2+1*3+5*-4 = 12+8+3+-20 = 3.<\/p>\n
\nformed by putting the tail of B to the tip of A, and then adding parallels for both A and B. C’s direction is perpendicular<\/em> to both A and B. There is one little trick – there are two<\/em> possible directions for the cross product A×B; which one to use is a matter of convention. The common convention that I was taught is called the right hand rule: take the pair of vectors, and put their tails together, so that they form a V. Then take your right hand<\/em>, with the fingers curled from A towards B, the direction that your thumb<\/em> is pointing is the direction of the cross-product.<\/p>\n\n
\n[4*6-7*5, 5*9-3*6, 3*7-4*9] = [24-35,45-18,21-36] = [-11,27,-15].<\/p>\n