# Thread: Vector Proof - Dot and Cross Products

1. ## Vector Proof - Dot and Cross Products

Hello everyone,

I am having some trouble with the following problem and would appreciate help. Thank you!

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16. For any vectors $\displaystyle \vec{a}, \vec{b}, \vec{c}$, show that $\displaystyle (\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \bullet \vec{c})\vec{b} - (\vec{b} \bullet \vec{c})\vec{a}$.

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I worked on the RS since it is more complicated:

RS = $\displaystyle (|\vec{a}||\vec{c}|\cos \theta)\vec{b} - (|\vec{b}||\vec{c}|\cos \theta)\vec{a})$

However, I do not know how to distribute the vectors $\displaystyle \vec{b}$ and $\displaystyle \vec{a}$ into the dot products.

2. Have you already proven this $\displaystyle a \times (b \times c) = \left( {a \cdot c} \right)b - \left( {a \cdot b} \right)c$?

3. Originally Posted by Plato
Have you already proven this $\displaystyle a \times (b \times c) = \left( {a \cdot c} \right)b - \left( {a \cdot b} \right)c$?

I have not proven what you wrote but I do understand it; $\displaystyle a \times (b \times c)$ = $\displaystyle (a \times b) \times c$ because the vector perpendicular to these vectors would be equal regardless of the order of the calculations.

How would I use your identity to simplifying the RS?

4. Actually $\displaystyle a \times (b \times c) \ne \left( {a \times b} \right) \times c$.
Because the LHS is a linear combination of $\displaystyle b\;\&\; c$.
The RHS is a linear combination of $\displaystyle a\;\&\; b$.

So as painful as it is the best way is to use coordinates for each vector.
It is a messy, messy exercise in subscripts and factoring.

5. Originally Posted by Plato
Actually $\displaystyle a \times (b \times c) \ne \left( {a \times b} \right) \times c$.
Because the LHS is a linear combination of $\displaystyle b\;\&\; c$.
The RHS is a linear combination of $\displaystyle a\;\&\; b$.

So as painful as it is the best way is to use coordinates for each vector.
It is a messy, messy exercise in subscripts and factoring.

Would you mind elaborating on $\displaystyle a \times (b \times c) \ne \left( {a \times b} \right) \times c$?