Vector Product Definition Help

Hi everyone (first post here)

I'm struggling to understand the first principles type definition of the vector product. I can't find anything anywhere on the internet that satisfies me. All explanations I find make assumptions. (A bit like saying e^x by definition differentiates to itself, or worse then says it must because of the Maclaurin expansion... which was derived from the previous fact!!!)

So some definitions use the formula $\begin{pmatrix} a_{1} \\ a_{2} \\ a_{3} \end{pmatrix} \times \begin{pmatrix} b_{1} \\ b_{2} \\ b_{3} \end{pmatrix} = \begin{pmatrix} a_{2}b_{3}-a_{3}b_{2} \\ a_{3}b_{1}-a_{1}b_{3} \\ a_{1}b_{2}-a_{2}b_{1} \end{pmatrix}$
Which is not helpful, I know where it comes from - the definitions of cross product for unit vectors (I am very happy with those, including handedness) - and this would be great, but I cannot find a proof that explains why vector products are/should be distributive across addition.

Other definitions jump straight to $\mathbf{a} \times \mathbf{b} = \mathbf{\hat{n}}|a||b|sin \theta$ with absolutely no justification.
Can someone please help me understand it from the ground up?

To a certain extent, the vector product is just made up - you could invent a definition of a vector product to suit yourself and see what conclusions follow from that definition.

I don't know if this approach will satisfy you either, but if you define the results of the vector product for the unit vectors i, j, k as ixj =k, jxk = i, kxi=j and accept that this is an antisymmetric function so that jxi = -k for example, then the general expression for the vector product of 2 vectors (a,b,c) and (d,e,f) follows when you write them in component form.

You missed my point about distributivity.

Are you asking why a x (b + c) = (a x b) + (a x c) when a, b and c are all vectors?

I'm not sure what you would accept as an explanation - I have a justification in a vector calculus textbook which isn't easy for me to transcribe into a post here but basically shows a triangle formed from the vectors b, c and b+c under the effect of the cross-product by a vector a which is equivalent to a scaling and rotation.

The book is "Vector Calculus" by P C Matthews in the Springer Undergraduate series if you're interested

Correct, thanks, I'll have a look, however if you or anyone else has a proof based on A-Level maths that would be nice

ut I cannot find a proof that explains why vector products are/should be distributive across addition.

Oh take no notice of the book series title - the "Undergraduate" simply means that the book starts from definitions and then works up to uni topics. The cross product definition and properties are given right at the start and I'm sure you would have no problem understanding it

As it happens, I've just tried googling "vector product" + "distributivity" and found a link to another book that takes exactly the same approach to explaining things - see this:

http://books.google.co.uk/books?id=CRIjIx2ac6AC&pg=PA17&lpg=PA17&dq=%22vector+product%22+%2B+%22distributivity%22&source=bl&ots=oHJHkMW5sS&sig=ODyXqqcAfAlUYGIPyzpKZ6uJiu8&hl=en&sa=X&ei=vaJpUdvXAuKJ0AXB7oDQDw&ved=0CFQQ6AEwBw#v=onepage&q=%22vector%20product%22%20%2B%20%22distributivity%22&f=false

If this link takes you to a preview screen of 3 results, click on the top image - I'm not sure if you can see the whole book, but you can definitely scroll up and down a few pages to see the bit you're interested in!

This follows from the properties of the scalar triple product $(a \times b) \cdot c \equiv [a,b,c] = [b,c,a] = [c,a,b]$: set $d=a \times (b+c) - a \times b - a\times c$, and consider $d \cdot d = d \cdot (a \times (b+c)) - d \cdot (a \times b) - d \cdot (a \times c) = d \cdot (a \times (b+c)) - (d \times a)\cdot b - (d \times a) \cdot c = d \cdot (a \times (b+c)) - (d \times a)\cdot (b+c)$ by the distributivity of the dot product, so $d\cdot d = d \cdot (a \times (b+c)) - d \cdot (a \times (b+c)) = 0$, so d = 0.

Thanks, thats really nice, as long as our initial definition is that axb=n*a*b*sinø, because otherwise the projection is a load of rubbish. You have more or less solved the problem, because with distributivity I can equate it to the vector formula.

Although my problem is solved, I would love to see a proof that uses only the definition of vector products for the unit vectors i, j and k. I mean that we derive the sine property.

I'm not sure that would be possible because a vector is defined by both a direction and a magnitude, so if we invent a concept of "vector product" that operates on 2 vectors and produces a vector as output, we need to be able to say (a) how that vector's direction is defined; (b) what the magnitude of that output vector should be?

I believe the "projection definition" satisfies those criteria and therefore specifies the vector product uniquely.

I understand your desire to get to the bottom of A level concepts wherever possible, but a lot of the objects we work with do just arise because someone at some point has come up with a definition and then developed it to see what the consequences are.

If you look at more abstract entities like groups, rings, fields, vector spaces etc, you'll see that they're basically just a list of objects and properties with various axioms and restrictions applied to them.

Sensible argument, thank you Can't wait to learn about those at university!

At least you're asking more challenging questions than most A level students


I take it you are studying for A levels at present?

Yes and I'm preparing for STEP II and III!! :P

And I have a weird thing where I can only learn by seeing derivations not blind definitions.

That's not weird in the slightest - anecdotal evidence suggests that almost everyone learns something better having seen it derived (me included!)

Well it's quite useful, because in theory I can't remember any incorrect mathematical facts