The following figure shows two vectors a and b at an angle θ extended to form a parallelogram.
From the above figure in ΔOMN,
sinθ= MN OM = MN | b | MN=| b |sinθ (1)
The magnitude for cross product of aand b is,
| a×b |=| a || b |sinθ
From the equation (1),
| a×b |=( OK )( MN )
Multiply and divide by 2.
| a×b |=( OK )( MN )× 2 2 =2×Area of ΔOMK Area of ΔOMK= 1 2 | a×b |
Thus, the area of the triangle contained between the vectors a and b is one half the magnitude of the a×b.
The resultant of two vectors A and B is perpendicular to the vector A and its magnitude is equal to half the magnitude of vector B. The angle between A and B is