Problem on a Vector Concept Question

Hey everyone, I have a conceptual question regarding vectors. The question states: Which of the following expressions are well defined for all vectors a, b, c, and d?

A. a * (b x c)

B. magnitude of (a) x (b x c)

C. (a*b) x (c*d)

At first glance, I recognize that (A) is a scalar and that (C) is a cross product of dot products, and (B) and I am not quite sure on (although it appears to be a vector). However, the question is throwing me off. What does it mean to be well defined for all vectors a,b,c, and d? Only (B) appears to be a vector, while (A) and (C) are not. A friend told me that they are all well defined, but I am not sure if he is correct or not. How exactly would I approach this problem?

Any thoughts?

Any feedback appreciated, thanks.

Re: Problem on a Vector Concept Question

Quote:

Originally Posted by

**Beevo** Hey everyone, I have a conceptual question regarding vectors. The question states: Which of the following expressions are well defined for all vectors a, b, c, and d?

A. a * (b x c)

B. magnitude of (a) x (b x c)

C. (a*b) x (c*d)

Well you have work yet to do.

$\displaystyle a\cdot(b\times c)$ is a scalar. Dot products give scalars.

$\displaystyle a\times(b\times c)=(a\cdot c)b-(a\cdot b)b$ so the magnitude is a scalar.

However, part C. is undefined because $\displaystyle \text{scalar}\times\text{scalar}$ is not defined.

Re: Problem on a Vector Concept Question

look at (C) again ... how is the cross product of scalar quantities defined?

Re: Problem on a Vector Concept Question

Quote:

Originally Posted by

**Plato** Well you have work yet to do.

$\displaystyle a\cdot(b\times c)$ is a scalar. Dot products give scalars.

$\displaystyle a\times(b\times c)=(a\cdot c)b-(a\cdot b)b$ so the magnitude is a scalar.

However, part C. is undefined because $\displaystyle \text{scalar}\times\text{scalar}$ is not defined.

Right, the third part is meaningless because you are multiplying two scalars. However, part A and B are both scalars, but the question asks which of the following are well defined for all vectors. Since there are no vectors, I assume it would be none of them.

Quote:

Originally Posted by

**skeeter** look at (C) again ... how is the cross product of scalar quantities defined?

It is not defined. You can't multiply a scalar x scalar, so it is meaningless I guess.

Re: Problem on a Vector Concept Question

Quote:

Originally Posted by

**Beevo** It is not defined. You can't multiply a scalar x scalar, so it is meaningless I guess.

Well you are confusing *vector products* with *scalar products*.

*Vector products* yield a vector.

* Scalar products* yield a scalar.