Dot Product

The scalar or dot product of the vectors V 1 ⃗ vec V _{1} and V 2 ⃗ vec V _{2} is given by

V 1 ⃗ . V 2 ⃗ = ‖ V 1 ⃗ ‖ . ‖ V 2 ⃗ ‖ . cos ( α ) vec V _{1}. vec V _{2}= ldline vec V _{1} rdline . ldline vec V _{2} rdline . cos( %alpha )

Such that α is the angle between the two vectors

Note

The scalar product of two vectors is equal to the product of the modulus of one of the vectors in the projection of the modulus of the other vector onto the carrier of this vector.

Analytical Expression of the Dot Product

We have the two vectors  V 1 ⃗ vec V _{1} and V 2 ⃗ vec V _{2} in the basis i ⃗ , j ⃗ , k ⃗ vec i , vec j , vec k with V 1 ⃗ = x 1 i ⃗ + y 1 j ⃗ + z 1 k ⃗ vec V _{1}= x_{1} vec i + y_{1} vec j+ z_{1} vec k and V 2 ⃗ = x 2 i ⃗ + y 2 j ⃗ + z 2 k ⃗ vec V _{2}= x_{2} vec i + y_{2} vec j+ z_{2} vec k

The dot product of these two vectors is expressed by: V 1 ⃗ . V 2 ⃗ = x 1 x 2 + y 1 y 2 + z 1 z 2 vec V _{1}. vec V _{2}= x_{1} x_{2}+ y_{1} y_{2}+ z_{1} z_{2}