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

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. As well, is C= the only combination, for example doe A = BxC or is a negative there somewhere or is it not possible?

Thanks.

Stu
Jul 26, 2014

#2**+13 **

*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 **A**x**B** 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 **B**x**A** is a vector of the same magnitude* but it points in the opposite direction*.

Your first image shows **r**x**F** is the vector **M _{0}**. In this case

Alan
Jul 27, 2014

#2**+13 **

Best Answer

*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 **A**x**B** 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 **B**x**A** is a vector of the same magnitude* but it points in the opposite direction*.

Your first image shows **r**x**F** is the vector **M _{0}**. In this case

Alan
Jul 27, 2014

#3**+10 **

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

Alan
Jul 28, 2014

#4**+5 **

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.

Stu
Jul 31, 2014

#5**+10 **

*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!

Alan
Jul 31, 2014

#7**+5 **

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.

Stu
Jul 31, 2014