How can I solve for a vector in a tensor equation involving dot products?

[zephyr5050]

Homework Statement

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?

Homework Equations

That's what I'm here for.

The Attempt at a Solution

No idea.

 

[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 = c⋅d ⇔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...