# Math Help - Problem on a Vector Concept Question

1. ## 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.

2. ## Re: Problem on a Vector Concept Question

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

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

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

3. ## Re: Problem on a Vector Concept Question

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

4. ## Re: Problem on a Vector Concept Question

Originally Posted by Plato
Well you have work yet to do.
$a\cdot(b\times c)$ is a scalar. Dot products give scalars.

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

However, part C. is undefined because $\text{scalar}\times\text{scalar}$ 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.

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.

5. ## Re: Problem on a Vector Concept Question

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.