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! :)