# Vector Product

• Nov 3rd 2010, 02:03 AM
davefulton
Vector Product
Dear all,

I have a vector identity and am confused about the notation

$\displaystyle \bf a\times (\bf b\times \bf c)=\bf b(\bf a\cdot \bf c)-\bf c(\bf a\cdot\bf b)$

where $\displaystyle \bf a$, $\displaystyle \bf b$ and $\displaystyle \bf c$ are vectors and $\displaystyle \times$ and $\displaystyle \cdot$ are the cross and dot product respectively.

My question is what is the operation between $\displaystyle \bf b(a$?

It isn't the dot product. Is it the scalar product? Also is it commutative?

$\displaystyle \bf b(\bf a\cdot \bf c)-\bf c(\bf a\cdot\bf b)=(\bf a\cdot \bf c)\bf b-(\bf a\cdot\bf b)\bf c$
• Nov 3rd 2010, 03:07 AM
tonio
Quote:

Originally Posted by davefulton
Dear all,

I have a vector identity and am confused about the notation

$\displaystyle \bf a\times (\bf b\times \bf c)=\bf b(\bf a\cdot \bf c)-\bf c(\bf a\cdot\bf b)$

where $\displaystyle \bf a$, $\displaystyle \bf b$ and $\displaystyle \bf c$ are vectors and $\displaystyle \times$ and $\displaystyle \cdot$ are the cross and dot product respectively.

My question is what is the operation between $\displaystyle \bf b(a$?

It isn't the dot product. Is it the scalar product? Also is it commutative?

$\displaystyle \bf b(\bf a\cdot \bf c)-\bf c(\bf a\cdot\bf b)=(\bf a\cdot \bf c)\bf b-(\bf a\cdot\bf b)\bf c$

The only possible, and sensical, meaning of $\displaystyle b(a\cdot c)$ is the vector $\displaystyle b$ multiplied by the scalar $\displaystyle a\cdot c$ ...Now you prove this.

Tonio

Pd. I'm assuming you're working with vectors in $\displaystyle \mathbb{R}^3$...?
• Nov 3rd 2010, 04:20 AM
davefulton
The dot product of any two vectors is a scalar. This answers my question. Thank you. I suppose this implies it is commutative also.
• Nov 3rd 2010, 05:02 AM
tonio
Quote:

Originally Posted by davefulton
The dot product of any two vectors is a scalar. This answers my question. Thank you. I suppose this implies it is commutative also.

Well, the product of a scalar and a vector is obviously commutative, as is the dot product, but NOT the cross product!

Tonio