Reciprocal System of Vectors

Projection of a vector on a line:

The projection of a vector on a line is the component of the vector in the direction of that line.

The vector OP is known as the projection vector of b on a, when,

Projection vector of 

Area of Parallelogram:

In the figure,  (altitude of parallelogram OACB)

⇒ | a × b | = area of parallelogram OACB

Hence the magnitude of the vector product is equal to the area of the parallelogram formed with a and b as adjacent sides.

The vector whose magnitude is equal to the area of the figure and whose direction is outwards (normal to the plane of the figure) is known as vector area of that figure.

⇒ vector area of the parallelogram 

Area of Triangle:

Vector Area = 

(Note that the area of triangle ABC is half the area of parallelogram whose adjacent sides lie on the triangle)

Area of a Quadrilateral:

Area (ABCD) = Area (ABC) + Area (ACD)

(Note carefully that both the cross products on RHS should be taken in such a manner that their direction is outwards i.e., they should correspond to anti-clockwise rotations)

⇒ Area of quadrilateral  | vector product of diagonals |