Geometric Interpretation of Cross Product of Real Vectors

1. The Length/Magnitude of the Resultant Vector of Cross Product between any 2 Real Vectors \(\vec{A}\) and \(\vec{B}\) can be Geometrically Interpreted in following 3 equivalent ways
   
   \(|\vec{A}\times\vec{B}|=|\vec{B}\times\vec{A}|=\) Length of \(\vec{A}\hspace{.2cm}\times \) Length of \(\vec{B}\hspace{.2cm}\times \) (Sine of Angle Between \(\vec{A}\) and \(\vec{B}\)) \(=|\vec{A}||\vec{B}| \sin (\theta)\)
   
   \(|\vec{A}\times\vec{B}|=|\vec{B}\times\vec{A}|=\) Length of Rejection of \(\vec{A}\) from \(\vec{B}\hspace{.2cm}\times\) Length of \(\vec{B}=(|\vec{A}|\sin (\theta))|\vec{B}| \)
   
   \(|\vec{A}\times\vec{B}|=|\vec{B}\times\vec{A}|=\) Length of Rejection of \(\vec{B}\) from \(\vec{A}\hspace{.2cm}\times\) Length of \(\vec{A}=(|\vec{B}|\sin (\theta))|\vec{A}| \)
   
   In the above equations \(\theta\) is the Angle Between \(\vec{A}\) and \(\vec{B}\) such that \(0 \leq \theta \leq 180^\circ\).
2. The Length/Magnitude of the Resultant Vector of Cross Product between Real Vectors \(\vec{A}\) and \(\vec{B}\) also gives the Area of the Parallelogram whose 2 Adjacent Sides are given by Vectors \(\vec{A}\) and \(\vec{B}\)