The formula for the magnitude of cross product, $\| u\times v \| = \| u \| \|v \| \sin \theta $ Can someone show me a proof of the magnitude (length) of the cross product: $$\|u \times v \| = \| u \| \|v \| \sin \theta $$
 A: It depends on the definition of cross product you use. The only definition for which the absolute value is not immediate is
$$
(x,y,z)\times(u,v,w)=(yw-zv,zu-xw,xv-yu)\tag{1}
$$
Note that the dot product is
$$
(x,y,z)\cdot(u,v,w)=xu+yv+zw\tag{2}
$$
A rotation is represented by a matrix $R$ so that $R^TR=I$. The dot product is invariant under rotations:
$$
\begin{align}
(Ra)\cdot(Rb)
&=(Ra)^T(Rb)\\
&=a^TR^TRb\\
&=a^Tb\\
&=a\cdot b\tag{3}
\end{align}
$$
Since lengths are computed using dot products, lengths are preserved by rotations, also. Thus, the side lengths of the triangle formed by $a$, $b$, and $b-a$ are preserved, and likewise, $\theta$, the angle between $a$ and $b$.
If $R$ rotates $a$ to the positive $x$-axis, then
$$
\begin{align}
a\cdot b
&=(|a|,0,0)\cdot(b_x,b_y,b_z)\\
&=|a|b_x\\
&=|a||b|\cos(\theta)\tag{4}
\end{align}
$$
$\hspace{3.5cm}$
The sum of the square of the length of the cross product and the square of the dot product is
$$
\begin{align}
&|(x,y,z)\times(u,v,w)|^2+((x,y,z)\cdot(u,v,w))^2\\
&=(yw-zv)^2+(zu-xw)^2+(xv-yu)^2+(xu+yv+zw)^2\\
&=(yw)^2+(zv)^2+(zu)^2+(xw)^2+(xv)^2+(yu)^2
+(xu)^2+(yv)^2+(zw)^2\\
&=x^2(u^2+v^2+w^2)+y^2(u^2+v^2+w^2)+z^2(u^2+v^2+w^2)\\
&=(x^2+y^2+z^2)(u^2+v^2+w^2)\\
&=|(x,y,z)|^2|(u,v,w)|^2\tag{5}
\end{align}
$$
Equations $(4)$ and $(5)$ yield
$$
\begin{align}
|(x,y,z)\times(u,v,w)|^2
&=|(x,y,z)|^2|(u,v,w)|^2-((x,y,z)\cdot(u,v,w))^2\\
&=|(x,y,z)|^2|(u,v,w)|^2-|(x,y,z)|^2|(u,v,w)|^2\cos^2(\theta)\\
&=|(x,y,z)|^2|(u,v,w)|^2(1-\cos^2(\theta))\\
&=|(x,y,z)|^2|(u,v,w)|^2\sin^2(\theta)\tag{6}
\end{align}
$$
Therefore,
$$
|a\times b|=|a||b||\sin(\theta)|\tag{7}
$$
A: The magnitude of the cross product is the positive area of the parallelogram having $a$ and $b$ as sides:

From here, it is easy to see that when the angle is 0 or 180 degrees, the area of the parallelogram would be 0. Similarly, the area is largest when the angle is 90 degrees.
