Vector Cross and Dot Products: Understanding and Solving Problems

[clementc]

Hey guys, I'm a kinda noobie to this site so I have not much experience with the formatting and stuff here, but anyway was doing some physics and came stuck =P Would really appreciate any help

Homework Statement

Vectors A and B are drawn from a common point, with the angle in between them \theta.
(a) What is the value of A \times (B \times A)?
Now consider any three vectors A, B and C:
(b) Prove that: A \times ( B \times C) = B( A \cdot C) - C( A \cdot B)
(c) Are the two products A \times ( B \times C) and ( A \times B) \times C equal in either magnitude or direction? Prove your answer.

Homework Equations

I think you would need to use
A \cdot B = \left|A\right| \left|B\right| \cos \theta
and that the magnitude of A \times B is \left|A\right| \left|B\right| \sin \theta
and the right hand rule of course

The Attempt at a Solution

I don't know =( I can do every question on this problem set except these parts
pleasepleaseplease help

 

[cathode-ray]

Hi!

This is how I did:

a)

Let:

\overrightarrow{A} = A \overrightarrow{e}_{A}

\overrightarrow{B}=B \overrightarrow{e}_{B}

\overrightarrow{e}_{B}\times\overrightarrow{e}_{A}=\overrightarrow{e}_{C}

\overrightarrow{e}_{A}\times\overrightarrow{e}_{C}=\overrightarrow{e}_{D}

\overrightarrow{A}\times(\overrightarrow{B}\times\overrightarrow{A})=\overrightarrow{A}\times(AB\sin(\theta)\overrightarrow{e}_{C})

As \overrightarrow{e}_{A} is perpendicular to \overrightarrow{e}_{C} the angle between them is \frac{\pi}{2} we get

=A^{2}B\sin(\theta)\overrightarrow{e}_{D}


b) & c) For these, one way is to write the vectors in component form. There is already a similar discussion about that:

https://www.physicsforums.com/showthread.php?t=352134

and you can see also:

http://en.wikipedia.org/wiki/Triple_product#Vector_triple_product


I hope this helps.

 

[clementc]

i don't really get the \overrightarrow{e} notation it seems really weird
but THANKYOUT THAKNK YOU for the triple vector product thingo - i had no idea it had a name but managed to find proofs for it once iknew the name
they way i just did it was brute expand LHS and RHS! nothing like a page of algebra bash xDD

 

[cathode-ray]

The \overrightarrow{e} is referring to the unit vectors of A,B,C,D. For example in the cartesian coordinate system it is used \overrightarrow{e}_{x}, \overrightarrow{e}_{y}, \overrightarrow{e}_{z}, each one related to one of the three axis (http://en.wikipedia.org/wiki/Standard_basis" ). In here I use them just to break each one of the vectors in its direction given by \overrightarrow{e} and its magnitude given by the name of the vector. This way the result can be generalized to any vector.

 

[clementc]

OH its the same as \hat{i}, \^{j}, \textrm{and}\ \^{k} isn't it? xD

 

[cathode-ray]

It's the same idea but in this case each one of the vectors \overrightarrow{e} have an individual combination of those vectors. In other words: \overrightarrow{e}=(a\hat{i},b\hat{j},c\hat{i}) such that ||\overrightarrow{e}||=1.

 

[cathode-ray]

Note: It is \overrightarrow{e}=(a\hat{i},b\hat{j},c\hat{k}), I forgot to change the last i.

 

[clementc]

oh haha yeah i see it =D
ty ty ty tyyyyyyyyyy thank you hehe
and MERRY CHRISTMASSSS EVE to you =]

 

[cathode-ray]

Thanks! Merry Christmas to you too, and a Happy New Year! :)