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.

Using property 5, we get $(a \times  b) \cdot (c \times d) = ((a \times  b) \times c) \cdot d$. Then use property 1 to get $((a \times  b) \times c) \cdot d = (-c \times (a \times  b)) \cdot d$. Now we can apply property 6 to get $(-c \times (a \times  b)) \cdot d = -((c\cdot b)a-(c\cdot a)b)\cdot d$. If you simplify this expression using linearity of the dot product, you should get your expansion for the determinant.
