Analytic geometry Scalar Triple Product.

[Alister]

Let a and b be non zero and non parallel vectors. Show that for any non zero vector c, the following three vectors a, b and (a x b) x c are coplanar.

 

[Deleted member 4993]

Let a and b be non zero and non parallel vectors. Show that for any non zero vector c, the following three vectors a, b and (a x b) x c are coplanar.

When 3 non-zero vectors ARE coplanar - What would be their scalar product?

 

[Alister]

When 3 non-zero vectors ARE coplanar - What would be their scalar product?

zero??

 

[Deleted member 4993]

zero??

Correct!

Please find out:

How can you prove - when 3 non-zero vectors ARE coplanar their scalar product is zero.

I would use similar procedure for the problem given to you.

 

[pka]

Let a and b be non zero and non parallel vectors. Show that for any non zero vector c, the following three vectors a, b and (a x b) x c are coplanar.

Suppose \(\left\{\vec{u},~\vec{v},~\vec{w}\right\}\) is a set of three vectors no two of which are parallel.
Then \(\vec{z}=\vec{u}\times(\vec{v}\times\vec{w})=(\vec{u}\cdot\vec{w}){\bf\vec{v}}-(\vec{u}\cdot\vec{v})\bf\vec{w}\).
That is a vector triple product. Please note the order in which the terms are written.
Moreover, note that the vector \(\bf\vec{z}\) is a linear combination of \(\bf\vec{v}~\&~\vec{w}\).
Also note that \(\bf\vec{u}\times(\vec{v}\times\vec{w})\ne(\vec{u}\times\vec{v})\times\vec{w}\) (i.e.) it is not associative.