How to use Cross Product Properites to do proof How do I proceed with a proof for this question?
Prove that:
\begin{equation}
       (a \times  b) \cdot (c \times d)  =  \begin{vmatrix}
    a \cdot c & b \cdot c \\
    a \cdot d & b \cdot d \\
    \end{vmatrix} 
\end{equation}
I have to use the cross product properties to do the proof:

So far I have taken the det of the right side:
\begin{equation}
(a \times b) \cdot (c \times d) = (a \cdot c) \cdot (b\cdot d) - (b \cdot c)(a \cdot d) 
\end{equation}
I don't understand how to use the properities after this step.

Property 5 states that:
$$\color{blue}{u}\cdot (\color{red}{v}\times\color{green}{w})=(\color{blue}{u}\times\color{red}{v})\cdot\color{green}{w}$$
Applying property 5 to the L.H.S. we get:
$\color{blue}{(a\times b)}\cdot (\color{red}{c}\times \color{green}{d}) = (\color{blue}{(a\times b)}\times \color{red}{c})\cdot \color{green}{d}$
Property 1 states that:
$$\color{blue}{u}\times \color{red}{v}=\color{red}{-v}\times\color{blue}{u}$$
So, applying property 1 to our previous step:
$(\color{blue}{(a\times b)}\times \color{red}{c})\cdot \color{green}{d}=(\color{red}{-c}\times \color{blue}{(a\times b)})\cdot \color{green}{d}$
Now that we have something of the form $u\times (v\times w)$, we can use property 6.
Continue the problem using property 6 to get rid of the $\times$ products in the expression and replace it with dot products, addition, and subtraction.  Continue simplifying to get to the result.

Property 6 states that:
$$\color{blue}{u}\times(\color{red}{v}\times\color{green}{w})=(\color{blue}{u}\cdot\color{green}{w})\color{red}{v}-(\color{blue}{u}\cdot\color{red}{v})\color{green}{w}$$
Using prop6 should only affect what is inside of the black parenthesis that I have written previously.  
$=(\color{blue}{-c}\times (\color{red}{a}\times\color{green}{b}))\cdot d = ((\color{blue}{-c}\cdot\color{green}{b})\color{red}{a}-(\color{blue}{-c}\cdot\color{red}{a})\color{green}{b})\cdot d$
