Vector addition and multiplication?

What are the differences when using sum of vectors and multiplication of vectors? 

When you sum two vectors you just add corresponding components.  There are two sorts of vector multiplication - your example refers to cross-multiplication.

Given that multiplication of vectors is not cummulative; is the order always the fist of x,y,z multiplied with the later?Why is it called cross multiplication? ie, cross multiplication C=AxB and C= AxBsin(Theta) and so on. It seems straight multiplication to me.

You are correct that cross-mutiplication is not commutative.   If A and B are vectors then AxB is also a vector. It has magnitude ABsin(θ), where θ is the angle between them and A and B represent the magnitudes of A and B respectively, but it points out of the plane that contains A and B (in fact it is normal to that plane). The result of BxA is a vector of the same magnitude but it points in the opposite direction.

Your first image shows rxF is the vector M₀.  In this case r and F are at right-angles to each other so sin(θ) = 1 and M₀ = rF.  To find the direction of M₀ use the right-hand rule.  Hold your (right hand's) thumb, index and middle fingers at right-angles to each other, point  your index finger in the direction of r, your middle finger in the direction of F and your thumb will be pointing in the direction of M₀.

Please give me a bit of a hand to understand this information in the below image. As you can see i've covered almost everything for week 1 now, and this is quite hard to interprut. Image source uni lectures, VU colaborate, 26/07/14. 

At the risk of confusing you still further, I've written a more detailed explanation of 3-d vectors below:

Thanks Alan. Yes, I understand a lot about the vectors but I didn't clearly know the difference between the cross multiplication and dot product/addition of forces. Here was much easier to ask than looking on Kahn. :)

We learnt this week, 3d vectors also solving with a matrix, and the information of the last 1/2 of your post. Your reply is now printed and a good reminder for this week at least while I wait for the text book to arrive. However the screen prints I posted were from last weeks lecture notes before we'd covered this more in depth. I guess that due to missing a lecture I just felt also that I may not have known the information while required knowing it to use in tutorial questions.I think most of the students felt this week a bit like the job was demanding given what we knew.

Additionally, I found that this week when trying to resolve the 3d vector equations, that I was trying to get a single answer in N rather than in the i, j, k form and leaving it in i, j, k. Thankfully it has become more clear now. This was the difference I needed to get my head around and which I was ultimately seeking somewhere within my search for answers, lol. Applying the cross multiplication solves for.. meaning what's at the root of it, what does it do exactly? r is lengths of the values of i j and x right, so then why do we use them to resolve magnitude? Also, if you do not have all the variables for the matrix will it generally come down to a quadratic or similtaneous eqaution which is solved by a matrix?

WE finished the tutorial this week on breaking down a similtaneous equation, into a quadradic and using the quadratic formula. So, we're catching up on the notes from last week as we dive a little deeper. Thanks again. It was not to confusing. It is just that the application is confusingas well as recalling all the information as required both that I have learnt recently or since the beginning of the course. It's becoming simpler though with a special thanks to the forum here and your self.

Applying the cross multiplication solves for.. meaning what's at the root of it, what does it do exactly?

Imagine you are in a spinning extra-vehicular harness in space (like George Clooney in the film Gravity).  You have to fire your pneumatic boosters to counteract the spin.  The reaction force from the gas jet is a vector, as is the radius arm from your centre of gravity to the gas jet.  The cross-product of these two vectors tells you the magnitude and direction of the torque being supplied to counter your spin (or to increase your spin, or set you tumbling in a different direction, if you get it wrong)!   

There are many more down-to-earth applications of the vector cross-product, often involving rotating objects.

 … if you do not have all the variables for the matrix will it generally come down to a quadratic or similtaneous equation which is solved by a matrix?

I don’t understand what you are asking here.  The matrix determinant form of the cross-product is just a way of remembering which components are multiplied by which in each direction! 

Hey Stu, if you are felling generous you could give Alan some thumbs up. 

So it all comes back to twist and/or rotate. I see more clear now. ... On another note, who's online who can help with a similataneous equation .... Yes more help .... using a matrix and 3 equations. I missed the class last week so not sure how to rearrange the equations into the form mx-ny=c. I'l give it a shot while I wait for a reply.