Vector Product

**davefulton** · November 3rd 2010, 02:03 AM

Dear all,

I have a vector identity and am confused about the notation

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

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

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

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

$ \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 $

**tonio** · November 3rd 2010, 03:07 AM

Originally Posted by davefulton

Dear all,

I have a vector identity and am confused about the notation

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

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

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

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

$ \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 $b(a\cdot c)$ is the vector $b$ multiplied by the scalar $a\cdot c$ ...Now you prove this.

Tonio

Pd. I'm assuming you're working with vectors in $\mathbb{R}^3$ ...?

**davefulton** · November 3rd 2010, 04:20 AM

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

**tonio** · November 3rd 2010, 05:02 AM

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

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