Vector Proof - Dot and Cross Products

Printable View

• Feb 25th 2009, 10:53 AM
scherz0
Vector Proof - Dot and Cross Products
Hello everyone,

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

---

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

---

I worked on the RS since it is more complicated:

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

However, I do not know how to distribute the vectors $\vec{b}$ and $\vec{a}$ into the dot products.
• Feb 25th 2009, 11:08 AM
Plato
Have you already proven this $
a \times (b \times c) = \left( {a \cdot c} \right)b - \left( {a \cdot b} \right)c$
?
• Feb 25th 2009, 11:14 AM
scherz0
Quote:

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

Thanks for your response, Plato.

I have not proven what you wrote but I do understand it; $
a \times (b \times c)$
= $(a \times b) \times c$ 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?
• Feb 25th 2009, 11:23 AM
Plato
Actually $a \times (b \times c) \ne \left( {a \times b} \right) \times c$.
Because the LHS is a linear combination of $b\;\&\; c$.
The RHS is a linear combination of $a\;\&\; b$.

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.
• Feb 25th 2009, 06:05 PM
scherz0
Quote:

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

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.

Thank you for your reply, Plato.

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