How to Prove the Vector Triple Product Identity?

In summary, the equation u x (v x w) = (u*w)v - (u*v)w can be proven using the \varepsilon_{ijk} notation and the summation convention. It is only necessary to prove it for three space, as the cross product is not defined for dimensions higher than three.
  • #1
robierob12
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0

Homework Statement




Prove that

u x (v x w) = (u*w)v - (u*v)w


Homework Equations



I've been trying to get this one but keep ending up no where.

I've tried the normal algebraic properties of the cross product but they lead me to a dead end.

What I am trying right now is just proving it in three space. Assigning each vector to a general form like (u1, u2, u3) and busting it out to see if I can get the right side identity.

Is there an easyier way to start this out.
Ideas are much appreciated.

Rob


The Attempt at a Solution

 
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  • #2
cross product isn't defined for dimensions higher than three, so you would just prove it for three space.
 
  • #3
I usually do these problems like these using the [itex]\varepsilon_{ijk}[/itex] thingy and the summation convention. If you're not familiar with that notation, this may just confuse you. The epsilon thingy is defined by [itex]\varepsilon_{ijk}=1[/itex] and the requirement that it's totally antisymmetric, i.e. that if you swap two indices, it will change sign. This implies for example that [itex]\varepsilon_{132}=-1[/itex] and that [itex]\varepsilon_{122}=0[/itex]. The "summation convention" is that I don't bother writing out the sigmas for summation, since all the indices that we need to sum over always occur exactly twice.

For example the scalar product [itex]u*v[/itex] is [itex]u_i v_i=v_1u_1+v_2u_2+v_3u_3[/itex] (I don't remember the LaTeX code for the scalar product) and the cross product [itex]u\times v[/itex] is [itex]\varepsilon_{ijk}u_jv_ke_i[/itex], where the [itex]e_i[/itex] are the basis vectors of [itex]\mathbb{R}^3[/itex].

The ith component of the left hand side of the equation you're trying to prove is by definition of the cross product

[tex](u\times (v\times w))_i=\varepsilon_{ijk} u_j (v\times w)_k=\varepsilon_{ijk}\varepsilon_{klm}u_j v_l w_m[/tex]
[tex]=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})u_j v_l w_m=u_j v_i w_j-u_j v_j w_i=(u*w)v_i-(u*v)w_i[/tex]

The tricky step is the one where I replaced two epsilons with some Kronecker deltas. The easiest way to see that this identity must hold is to explicitly calculate e.g. [itex]\varepsilon_{k12}\varepsilon_{k12}[/itex] and [itex]\varepsilon_{k12}\varepsilon_{k13}[/itex]. When you've done that, you'll probably understand.
 
Last edited:
  • #4
Thanks... I don't know why I was thinking of tring to prove it for vectors outside of three space.
 

1. What is a vector triple product?

A vector triple product is a mathematical procedure used to calculate the cross product of three vectors in three-dimensional space. It is typically represented as (A x B) x C.

2. How is the vector triple product calculated?

The vector triple product is calculated by first finding the cross product of the first two vectors, and then finding the cross product of that result with the third vector. This can be represented as (A x B) x C.

3. What is the geometric interpretation of the vector triple product?

The geometric interpretation of the vector triple product is that it represents the volume of a parallelepiped formed by the three vectors. This volume can be positive, negative, or zero, depending on the orientation of the vectors.

4. Is the vector triple product commutative?

No, the vector triple product is not commutative. This means that changing the order of the vectors in the product will result in a different value. (A x B) x C does not equal (B x A) x C.

5. What is the significance of the vector triple product in physics?

The vector triple product has many applications in physics, particularly in mechanics and electromagnetism. It is used to calculate torque, angular momentum, and magnetic fields. It also has significance in understanding the geometry and orientation of three-dimensional systems.

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