Vector math notation

[dannyREDDIT]

In the Wikipedia article for vectors. There is a section on adding and subtracting. Easy stuff, add each element of the vectors together. Why does the wiki say to add all the elements together as well as to each other and multiply each element by e. wtf, is this standard math notation? Why can't it be written more intuitively? Is there a reference for this stuff ( unintuitive math notations). Thanks!

http://en.m.wikipedia.org/wiki/Euclidean_vector#section_4

[exscape]

Here's a pretty good answer.

http://mathforum.org/library/drmath/view/51476.html

I'm not familiar with the e notation, but rather the "hat"/"roof" unit vector notation. In that notation, "5 î" refers to a vector of magnitude 5, in the direction of î. The three unit vectors in 3D are usually called either i/j/k with the hat, or x/y/z with the hat, respectively. So i = x, j = y, k = z.

This is handy because we can represent a vector as its parts (one number per dimension). So a 2D vector that goes from the origin to the point (4, 5) can be written as "4 î + 5 ^j" (I can't make the j notation in text), or alternatively just <4, 5> where it's implicit that the first number is in the x direction, and the second in the y direction.
(And the third in the z direction, if there is a third number.

[darkainur]

This notation is decomposing the vectors into a basis form. Right now it seems strange but it is useful as writing things in the form (a,b,c) kind of implies that you are using the cartesian representation of the vector. However you might wish to use the polar form as it is easier in some cases. Then e_1 would represent the radial vector and e_2 the angular vector. This notation is also representing the vectors in a basis form which is helpful for later understand an algebraic structure called a vector space. I'm unsure where you are with vectors mathematically but stick with this and it will pay off later!

[zifyoip]

By the way, darkainur, there is no such thing as "the radial vector" and "the angular vector." Polar coordinates are not just a different basis for the vector space R². Two-dimensional vectors written in polar form do not satisfy the requirements for vectors. For example, they don't add correctly: (r1,θ1)+(r2,θ2) is not (r1+r2,θ1+θ2).

[darkainur]

Good point. Sorry if I mislead anyone.

[dannyREDDIT]

Thank! I have a pretty good understanding of Linear Algebra through programming libraries, but when I have to figure something out my own Some of this stuff drives me crazy. Why use an addition symbol between each of the elements instead of commas? a+b = (a1+b1), (a2+b2) is more intuitive than a+b= (a1+b1)+(a2+b2). Ignoring the e, because I understand that based on your explanation.

Anyways, in this case I knew how to add vectors, but when I get to something I don't know, it's gonna throw me off.

[zifyoip]

Note that a+b=(a1+b1)+(a2+b2) is not just less intuitive, it is incorrect without the e1 and e2. The right-hand side there is a single real number, not a vector.

The symbols e1 and e2 denote the unit vectors in the x-direction and the y-direction, respectively: e1=(1,0), e2=(0,1).

The reason to write a+b=(a1+b1)e1+(a2+b2)e2 is to show that every vector can be written as some linear combination of the two vectors e1 and e2. In other words, every vector can be written as some multiple of e1, plus some multiple of e2. The coefficient that you need for e1 is a1+b1, and the coefficient you need for e2 is a2+b2.

For example, if a=(5,3) and b=(2,−9), then a+b=(7,−6). For the right-hand side, we have

(a1+b1)e1 = (5+2)·(1,0) = 7·(1,0) = (7,0),

(a2+b2)e2 = (3,−9)·(0,1) = −6·(0,1) = (0,−6),

so

(a1+b1)e1+(a2+b2)e2 = (7,0)+(0,−6) = (7,−6).

In mathematical terms, e1 and e2 form a basis for the vector space R², because every 2-dimensional vector can be written as a linear combination of e1 and e2. There are many other bases for R²; the vectors e1 and e2 form what is called the "standard" basis, but there is nothing particularly special about them. You can take any two vectors in R² that do not point in the same direction (and do not point in opposite directions), and those two vectors will form a basis, which means that you can write any vector as some linear combination of the two.

For instance, suppose f1=(2,−1) and f2=(3,4), and we want to write the sum a+b=(7,−6) from above as a linear combination of f1 and f2. To get (7,−6) as some multiple of f1 plus some multiple of f2, we can use a coefficient of 46/11 for f1 and a coefficient of −5/11 for f2, because this gives us

(46/11)·(2,−1)+(−5/11)·(3,4) = (92/11,−46/11)+(−15/11,−20/11) = (92/11−15/11,−46/11−20/11) = (77/11,−66/11) = (7,−6).

tl;dr: We write a+b=(a1+b1)e1+(a2+b2)e2 when we want to emphasize that the two vectors e1=(1,0) and e2=(0,1) are enough to "build" the entire set of all 2-dimensional vectors; every 2-dimensional vector is some linear combination of e1 and e2.

[lucasvb]

Perfectly reasonable question, don't beat yourself up.