=== Bevis ===
For krydsproduktet mellem <math>\vec{b}</math> og <math>\vec{c}</math> gælder:
:<math>\vec{b} \times \vec{c} =
\begin{pmatrix}
b_2  c_3 - b_3  c_2 \\
b_3  c_1 - b_1  c_3 \\
b_1  c_2 - b_2  c_1
\end{pmatrix}</math>
Derefter kan <math>\vec{a}</math> krydses på:
:<math>\begin{align}\vec{a} \times \left(\vec{b} \times \vec{c}\right) & =
\begin{pmatrix}
a_1 \\
a_2 \\
a_3
\end{pmatrix}\times \begin{pmatrix}
b_2  c_3 - b_3  c_2 \\
b_3  c_1 - b_1  c_3 \\
b_1 c_2 - b_2  c_1
\end{pmatrix}\\
\vec{a} \times \left(\vec{b} \times \vec{c}\right) & =
\begin{pmatrix}
a_2 b_1 c_2 - a_2 b_2 c_1 - a_3 b_3 c_1 + a_3 b_1 c_3\\
a_3 b_2  c_3 - a_3 b_3  c_2 - a_1 b_1 c_2 + a_1 b_2  c_1\\
a_1 b_3  c_1 - a_1 b_1  c_3 - a_2 b_2  c_3 + a_2 b_3  c_2
\end{pmatrix}\end{align}</math>
Det ses, at alle led er blandede, så et ublandet led
:<math>0= a_i b_i c_i - a_i b_i c_i</math>
lægges til og trækkes fra hver komponent:
:<math>\vec{a} \times \left(\vec{b} \times \vec{c}\right)  =
\begin{pmatrix}
a_2 b_1 c_2 - a_2 b_2 c_1 - a_3 b_3 c_1 + a_3 b_1 c_3 + a_1 b_1 c_1 - a_1 b_1 c_1\\
a_3 b_2  c_3 - a_3 b_3  c_2 - a_1 b_1 c_2 + a_1 b_2  c_1 + a_2 b_2 c_2 - a_2 b_2 c_2\\
a_1 b_3  c_1 - a_1 b_1  c_3 - a_2 b_2  c_3 + a_2 b_3  c_2 + a_3 b_3 c_3 - a_3 b_3 c_3
\end{pmatrix}</math>
Udtrykket kan nu opdeles i positive og negative led:
:<math>\vec{a} \times \left(\vec{b} \times \vec{c}\right)  =
\begin{pmatrix}
a_2 b_1 c_2 + a_3 b_1 c_3 + a_1 b_1 c_1\\
a_3 b_2  c_3 + a_1 b_2  c_1 + a_2 b_2 c_2\\
a_1 b_3  c_1  + a_2 b_3  c_2 + a_3 b_3 c_3
\end{pmatrix}-
\begin{pmatrix}
a_2 b_2 c_1 + a_3 b_3 c_1 + a_1 b_1 c_1\\
a_3 b_3  c_2 + a_1 b_1 c_2 + a_2 b_2 c_2\\
a_1 b_1  c_3 + a_2 b_2  c_3 + a_3 b_3 c_3
\end{pmatrix}</math>
Dette kan omarrangeres til at ses ud som BAC-CAB-reglen:
:<math>\begin{align}\vec{a} \times \left(\vec{b} \times \vec{c}\right) & =
\begin{pmatrix}
b_1(a_2 c_2 + a_3 c_3 + a_1 c_1)\\
b_2(a_2 c_2 + a_3 c_3 + a_1 c_1)\\
b_3(a_2 c_2 + a_3 c_3 + a_1 c_1)
\end{pmatrix}-
\begin{pmatrix}
c_1(a_2 b_2  + a_3 b_3  + a_1 b_1 )\\
c_2(a_2 b_2  + a_3 b_3  + a_1 b_1 )\\
c_3(a_2 b_2  + a_3 b_3  + a_1 b_1 )
\end{pmatrix}\\
\vec{a} \times \left(\vec{b} \times \vec{c}\right) & =
\begin{pmatrix}
b_1\\
b_2\\
b_3
\end{pmatrix}(a_1 c_1 + a_2 c_2 + a_3 c_3)-
\begin{pmatrix}
c_1\\
c_2\\
c_3
\end{pmatrix}(a_1 b_1 + a_2 b_2 + a_3 b_3)\\
\vec{a} \times \left(\vec{b} \times \vec{c}\right) & =\vec{b}\left(\vec{a} \cdot \vec{c}\right)-\vec{c}\left(\vec{a} \cdot \vec{b}\right)\end{align}</math>
Matematik Naturvidenskab