In this case the sum of their momentae add up to zero, I guess.
This would be the additive identity vector, I assume.
Ouch!
I had a whole bunch of questions, and as soon as I closed the window I realized I should have copied it in case my OP was consumed.
Anyway, I’ll try again:
Whats the deal with i, j and k in terms of vectors. I know they show directions but:
- Are they always perpendicular to eachother?
- Do they always correspond to certain directions?
- Why are they used as opposed to describing vectors in terms of x, y and z axises?
- What does a vector described in these terms, such as 2i + 2j mean?
- Whats the difference between a vector cross-product a dot-product, in non-mathematical terms?
- When/how does one use the right-hand rule in crossproducts or dot products?
- How does one solve cross-products and dot-products when i, j and k are used? Such as
2i X 2j and 2i dot 2j.
7a) How does one make a “dot” symbol on a keyboard? - How/When/Why does the order of multiplication matter in such problems?
Upon review, I realize that I’ve just asked more questions than I’d posted originally.
Any help would be appreciated. Thanks in advance.
- Are they always perpendicular to eachother?
Yes.
- Do they always correspond to certain directions?
i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1)
- Why are they used as opposed to describing vectors in terms of x, y and z axises?
They’re the same thing.
- What does a vector described in these terms, such as 2i + 2j mean?
ai + bj + ck = (a, b, c).
- Whats the difference between a vector cross-product a dot-product, in non-mathematical terms?
A dot product is defined for any two vectors in the same space, and results in a scalar (a real number). A cross product is only defined for vectors in R[sup]3[/sup], and gives you a vector perpendicular to the inputs.
- When/how does one use the right-hand rule in crossproducts or dot products?
Not sure. That’s a physics question, I think.
- How does one solve cross-products and dot-products when i, j and k are used? Such as
2i X 2j and 2i dot 2j.
2i dot 2j = 2 * 0 + 0 * 2 + 0 * 0 = 0. 2i X 2j is either 2k or -2k, although I’m not sure which. I can’t compute cross products in my head.
7a) How does one make a “dot” symbol on a keyboard?
Use (a, b)–this is another standard notation.
- How/When/Why does the order of multiplication matter in such problems?
The dot product is commutative. a X b = -(b X a), and the right-hand rule has something to do with that, although I’m not sure what.
Thanks, ultrafilter but a few questions about your answers:
What is R[sup]3[/sup]?
How did you calculate that dot product? The method, rather than the answer is what I need.
Also, for everyone else, I still need to know what to do with my hands. (see question 6.)
R[sup]3[/sup] is the set of vectors (a, b, c), where a, b, and c are real numbers.
For two vectors a = (a[sub]1[/sub], …, a[sub]n[/sub]) and b = (b[sub]1[/sub], …, b[sub]n[/sub]), (a, b) = a[sub]1[/sub]b[sub]1[/sub] + … + a[sub]n[/sub]b[sub]n[/sub].
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I’m not sure if you realize this or not, but another way to phrase ultrafilter’s answers to 1-3 is simply: the vector i is defined to be a vector of length one in the x-direction; j and k are likewise defined as as vectors of length one in the y- and z-direction, respectively. The i, j, and k notation is used to seperate the vector from the direction.
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The right-hand rule is not used in dot-products. It is used in a cross-product: The cross-product of two vectors is perpendicular to the plane formed by those two vectors. To determine which side of the plane the cross-product vector points, use the right-hand rule. In the case of AXB, curl the fingers of your right hand from the tip of A toward the tip of B. Your extended thumb points in the direction of the vector AXB.
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An easy way to solve cross-products: {x[sub]1[/sub]i + y[sub]1[/sub]j + z[sub]1[/sub]k} X {x[sub]2[/sub]i + y[sub]2[/sub]j + z[sub]2[/sub]k} can be solved by taking the determinate of this matrix:
|**i** **j** **k**|
|x[sub]1[/sub] y[sub]1[/sub] z[sub]1[/sub]|
|x[sub]2[/sub] y[sub]2[/sub] z[sub]2[/sub]|
The determinate will be a vector.
Note also that the right-hand rule is applied in right-hand vector spaces, where iXj = k. In other words, curl the fingers of your right hand from (1,0,0) toward (0,1,0). Your thumb will point along k.
umm, ultrafilter, 2ix2j = 4k, not 2k… 
The best way to compute crossproducts is as zut mentioned, with the determinantal approach. That saves having to memorize the (only slightly more complicated) general formula. The way I do the right hand rule is point my thumb in the direction of the first vector and my index finger in the direction of the second. Then an arrow pointing out from my palm is pointing in the direction of the crossproduct.
As a minor addition note the following (as zut mentioned):
ixj = k
From this and two other things (cyclic permutation and the fact that axb = -bxa), , everything follows. That is:
ixi = jxj = kxk = 0
ixj = k = -jxi
jxk = i = -kxj
kxi = j = -ixk
But…but…I like the distributive law!
Determinants are wonderful for finding cross products given components in any three orthogonal basis vectors.
It’s easy, though, to get the sign right for cross products. Just list off the vectors in modular order in your head. E.g… ijkijkijk. Now find which basis you start with go to the next basis. The very next one in that direction will be your answer and the direction you’re travelling in (as on the numberline) will be the sign of the answer. Remember that the order matters when taking cross products as g8rguy showed.
Since cross products distribute, you can go out and impress your friends with the ability to give cartesian cross products in your head. Though it’s sometimes a bit difficult to keep track when your given two three-component vectors, the accolades you’ll receive from people who know what you’re talking about will more than make up for it.
Background: I somehow managed to get through Algebra 1&2, Trig, AB Calculus in High School, and 2 additional semesters of College Calculus without ever learning linear algebra. I bought a book on it, but for some reason I’m not absorbing it.
So, my question: What are some applications? Where would you use it, and what calculations are simplified by using this instead of something else?
What fields would benefit most from learning this? (This is just a curiosity question. I fully intend to learn it regardless, just because I like learning math)
Well, dot products and cross products show up all over the place in vector calculus, of course (the divergence is a dot product and the curl is a cross product). This means that understanding this sort of thing is pretty important for dealing with Maxwell’s equations, to give what I would consider the most important example.
If you mean instead how is using the determinantal form to find the crossproduct advantageous, the answer is mostly that it’s a good mnemonic. I don’t use it myself.
OK, I’m only in high school, as I’d explained in my OP, which was eaten. So, how does one do matrices, (or if its too big for a post, a website would be appreciated.)
JSPrinceton: I dont understand your ijkijkijk trick. What are you considering a basis?
Well, knowing that the cross-product of two vectors is the determinant of a matrix (determinant, not determinate! I always screw that up!) is only a short-cut if you already know how to take a determinant. Otherwise, just remember the formula:
{x[sub]1[/sub]i + y[sub]1[/sub]j + z[sub]1[/sub]k} X {x[sub]2[/sub]i + y[sub]2[/sub]j + z[sub]2[/sub]k} = (y[sub]1[/sub]z[sub]2[/sub] - z[sub]1[/sub]y[sub]2[/sub])i + (z[sub]1[/sub]x[sub]2[/sub] - x[sub]1[/sub]z[sub]2[/sub])j + (x[sub]1[/sub]y[sub]2[/sub] - y[sub]1[/sub]x[sub]2[/sub])k
Note the nice symmetric structure.
JS Princeton’s ijkijkijk trick is a way to remember the answer to all the combinations of kXi or iXj, etc. A “basis” is one of the vectors i or j or k. To find the answer to iXk, for example, look at the string ijkijkijk. The answer to iXk is the next vector in the series (j), and, since you’re traveling to the left, it’s negative. Thus, iXk = -j.
Just to make life complicated, I note that it is possible to define basis vectors that are not mutually perpendicular. You might do this out of sheer perversity, or just for the fun of it, but such non-orthogonal systems have their uses. In crystal geometries, for instance, where the basis vectors correspond to the crystal axes. It’s usually a good idea to match the symmetry of your mathematical framework to that of your physical system.
I DARE you to learn electromagnetism in depth without dot products and cross products.
DARE you.
Note the ijkijkijk trick works with ANY ordered group of orthogonal bases. For example, in E&M spherical coordinates are often considered the most useful set of orthogonal bases in 3d. Knowing the order (r,phi,theta) or (rho,theta,phi) (depending on who you talk to) allows you to take trivial cross products in your head without twisting your wrists in weird ways. I swear, they should sell tickets to watch the Intro Physics course’s magnetism exam. They look SO FUNNY using the right hand rule. Then there’s always that one sorry soul in the corner who slept through class and crammed the night before, vainly employing their left hand and getting the absolute wrong answer (usually they’re not cleverly compensating for the charge carrier having a negative sign).
I should note that cross products are most useful in the real world since the real world is three dimensional and the cross product itself is inherently three dimenstional. There is a generalization of cross products into n-dimensions called Cartesian Products… but methinks that’s a different story for a different thread.
By the by, linear algebra is exceedingly useful in physics. For example, one of the clearest ways (IMHO) to understand the math behind quantum mechanics is almost entirely based on linear algebra.
I’ve done a bit of work with Cartesian products, and I don’t see how they’re a generalization of cross products. Could you explain?