The scalar product of two vectors A and B is equal to the magnitude of vector A times the projection of B onto the direction of vector A .

Consider two vectors A and B, the angle between them is .

Work, Energy and Power Commutative Law for Dot Product

Where BA represents the projection of vector B onto the direction of vector A .

Work, Energy and Power Commutative Law for Dot Product

Where AB represents the projection of vector A onto the direction of B vector .

Similarly,

Work, Energy and Power Commutative Law for Dot Product

This shows that the dot product of two vectors does not change with the change in the order of the vectors to be multiplied. This fact is known as the commutative of dot product.

Work, Energy and Power Commutative Law for Dot Product

According to distributive law for dot product:

Consider three vectors A, B and C .Geometric interpretation of dot product by drawing projection (Fig). First we obtain the sum of vectors B and C by head to tail rule then we draw projection OCA and ORA from the terminal point of vector (B+C) respectively onto the direction of A .

Work, Energy and Power Distributive Law for Dot Product

Work, Energy and Power Distributive Law for Dot Product

.( ) ( )

( )

( ) ( )

Pr

Pr

.( ) . .

A

A A A

A A A

A A

A

From figure

A B C A OR

A C R OC

A C R A OC

but

C R ojection of Bon A

C ojectionof C on A

Thus

A B C A B AC

The dot product A . (B + C) is equal to the projection of vector (B + C) onto the direction of A multiplied by the magnitude of A . i.e.