Geometric Interpretation of Dot/Scalar/Inner Product of Real Vectors

1. Dot/Scalar/Inner Product between any 2 Real Vectors \(\vec{A}\) and \(\vec{B}\) can be Geometrically Interpreted in following 3 equivalent ways
   
   \(\vec{A}\cdot\vec{B}=\) Length of \(\vec{A}\hspace{2mm}\times \) Length of \(\vec{B}\hspace{2mm}\times \) (Cosine of Angle Between \(\vec{A}\) and \(\vec{B}\)) \(=|\vec{A}||\vec{B}| \cos (\theta)\)   ...(1)
   
   \(\vec{A}\cdot\vec{B}=\) Length of Projection of \(\vec{A}\) on \(\vec{B}\hspace{2mm}\times\) Length of \(\vec{B}=(|\vec{A}|\cos (\theta))|\vec{B}| \)   ...(2)
   
   \(\vec{A}\cdot\vec{B}=\) Length of Projection of \(\vec{B}\) on \(\vec{A}\hspace{2mm}\times\) Length of \(\vec{A}=(|\vec{B}|\cos (\theta))|\vec{A}| \)   ...(3)
   
   In the above equations \(\theta\) is the Angle Between \(\vec{A}\) and \(\vec{B}\).
2. If Dot/Scalar/Inner Product between 2 Real Vectors is 0, that means the Angle between the 2 Vectors is \(90^{\circ}\) (i.e. they are Perpendicular/Orthogonal to each other).
   
   If Dot/Scalar/Inner Product between 2 Real Vectors is Greater Than 0,that means the Angle between the 2 Vectors is Acute (i.e. \(\geq 0^{\circ} and < 90^{\circ}\)).
   
   If Dot/Scalar/Inner Product between 2 Real Vectors is Lesser Than 0,that means the Angle between the 2 Vectors is Obtuse (i.e. \( > 90^{\circ} and \leq 180^{\circ}\)).