Vector Cross and Dot Products: Understanding and Solving Problems

In summary, the equation A \times (B \times C) = (B(A \cdot C) - C(A \cdot B)) can be solved for B and C using the right hand rule. Additionally, the products A \times (B \times C) and (A \times B) \times C are equal in magnitude but not in direction.
  • #1
clementc
38
2
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 [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.

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

The Attempt at a Solution


I don't know =( I can do every question on this problem set except these parts
pleasepleaseplease help
 
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  • #2
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.
 
  • #3
i don't 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
 
  • #4
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:
  • #5
OH its the same as [tex]\hat{i}, \^{j}, \textrm{and}\ \^{k}[/tex] isn't it? xD
 
  • #6
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].
 
  • #7
Note: It is [tex] \overrightarrow{e}=(a\hat{i},b\hat{j},c\hat{k}) [/tex], I forgot to change the last i.
 
  • #8
oh haha yeah i see it =D
ty ty ty tyyyyyyyyyy thank you hehe
and MERRY CHRISTMASSSS EVE to you =]
 
  • #9
Thanks! Merry Christmas to you too, and a Happy New Year! :)
 

1. What is a vector cross product?

A vector cross product is a mathematical operation that results in a vector that is perpendicular to both of the original vectors being multiplied. It is also known as the vector product or the cross product.

2. How is the cross product calculated?

The cross product of two 3-dimensional vectors, A and B, is calculated using the following formula: A x B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx).

3. What is the physical significance of the cross product?

The cross product has several physical interpretations, including determining the direction of a torque or moment of force, calculating the area of a parallelogram, and finding the direction of a magnetic field in a wire.

4. What is a vector dot product?

A vector dot product is a mathematical operation that results in a scalar (a single number) by multiplying the magnitudes of two vectors and the cosine of the angle between them. It is also known as the scalar product or the dot product.

5. How is the dot product calculated?

The dot product of two vectors, A and B, is calculated using the following formula: A · B = |A||B|cosθ, where |A| and |B| are the magnitudes of the vectors and θ is the angle between them. This can also be written as the sum of the products of the corresponding components: A · B = AxBx + AyBy + AzBz.

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