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Vector Cross and Dot Products

  • Thread starter clementc
  • Start date
38
0
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
1. Homework Statement
Vectors A and B are drawn from a common point, with the angle in between them [tex]\theta[/tex].
(a) What is the value of [tex] A \times (B \times A)[/tex]?
Now consider any three vectors A, B and C:
(b) Prove that: [tex] A \times ( B \times C) = B( A \cdot C) - C( A \cdot B)[/tex]
(c) Are the two products [tex] A \times ( B \times C)[/tex] and [tex]( A \times B) \times C[/tex] equal in either magnitude or direction? Prove your answer.

2. Homework Equations
I think you would need to use
[tex]A \cdot B = \left|A\right| \left|B\right| \cos \theta [/tex]
and that the magnitude of [tex]A \times B[/tex] is [tex]\left|A\right| \left|B\right| \sin \theta [/tex]
and the right hand rule of course

3. The Attempt at a Solution
I don't know =( I can do every question on this problem set except these parts
plzplzplz help
 
Hi!

This is how I did:

a)

Let:

[tex] \overrightarrow{A} = A \overrightarrow{e}_{A} [/tex]

[tex] \overrightarrow{B}=B \overrightarrow{e}_{B} [/tex]

[tex] \overrightarrow{e}_{B}\times\overrightarrow{e}_{A}=\overrightarrow{e}_{C} [/tex]

[tex] \overrightarrow{e}_{A}\times\overrightarrow{e}_{C}=\overrightarrow{e}_{D} [/tex]

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

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

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


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.
 
38
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i dont really get the [tex]\overrightarrow{e}[/tex] 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
 
The [tex] \overrightarrow{e} [/tex] is referring to the unit vectors of [tex] A,B,C,D [/tex]. For example in the cartesian coordinate system it is used [tex] \overrightarrow{e}_{x}[/tex], [tex] \overrightarrow{e}_{y}[/tex], [tex] \overrightarrow{e}_{z}[/tex], each one related to one of the three axis (http://en.wikipedia.org/wiki/Standard_basis" [Broken]). In here I use them just to break each one of the vectors in its direction given by [tex] \overrightarrow{e}[/tex] and its magnitude given by the name of the vector. This way the result can be generalized to any vector.
 
Last edited by a moderator:
38
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OH its the same as [tex]\hat{i}, \^{j}, \textrm{and}\ \^{k}[/tex] isn't it? xD
 
It's the same idea but in this case each one of the vectors [tex]\overrightarrow{e}[/tex] have an individual combination of those vectors. In other words: [tex]\overrightarrow{e}=(a\hat{i},b\hat{j},c\hat{i})[/tex] such that [tex]||\overrightarrow{e}||=1[/tex].
 
Note: It is [tex] \overrightarrow{e}=(a\hat{i},b\hat{j},c\hat{k}) [/tex], I forgot to change the last i.
 
38
0
oh haha yeah i see it =D
ty ty ty tyyyyyyyyyy thank you hehe
and MERRY CHRISTMASSSS EVE to you =]
 
Thanks! Merry Christmas to you too, and a Happy New Year! :)
 

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