Homework Help: Vector and Tensor Exercise

1. Mar 21, 2015

zephyr5050

1. The problem statement, all variables and given/known data

I'm currently trying to work through some issues I'm having with tensor and vector analysis. I have an equation of the form
$$\textbf{a} \bullet \textbf{b} = \textbf{c} \bullet \textbf{d}$$
where all quantities here are vectors. I want to solve for $\textbf{b}$ by finding an equation of the form
$$\textbf{b} = \overline{\textbf{T}} \bullet \textbf{d}$$
where $\overline{\textbf{T}}$ is a tensor. However I'm not sure the proper mathematical procedure to go about this. Any suggestions?

2. Relevant equations

That's what I'm here for.

3. The attempt at a solution

No idea.

2. Mar 21, 2015

Goddar

Hi. You can't possibly do that without imposing more constraints on your vector:
In tensor language, the vectors are contracted on both sides so you can't "solve" for b.
This may be confusing because a⋅b looks like a vector expression but it's really a scalar; if you want to solve for a vector you need a vector expression.
Look at the simplest example in 2 dimensions:
a⋅b = cd ⇔a1b1 + a2b2 = c1d1 + c2d2
You see that you would need two equations to solve for the two variables b1 and b2 , and for every additional dimension you need an additional constraint...