## Also Known as the Dot Product

The scalar product is one of two ways two of multiplying two vectors together. This method gives, as the name suggests, a scalar result.**A**.

**B**= |

**A**||

**B**|cos(ϑ)

**A**multiplied by the component of

**B**that is parallel to

**A**.

Clearly, from this definition, the dot products between the basis vectors in Cartesian coordinates (**i,j** and **k**) are given as:

**i**.

**i**= 1,

**i**.

**j**= 0,

**i**.

**k**= 0

**j**.**i** = 0, **j**.**j** = 1, **j**.**k** = 0

**k**.**i** = 0, **k**.**j** = 0, **k**.**k** = 1

**e**.

_{a}**e**= δ

_{b}_{ab}

**e**=

_{a}**i,j**or

**k**(for a = 1,2 or 3) and δ

_{ab}is the Kronecker delta (equal to one where a is equal to b zero otherwise).

This allows us to write the dot product between two vectors in terms of their Cartesian coordinates – avoiding the basically useless calculation of angles:

- (A

_{x}, A

_{y}, A

_{z}).(B

_{x}, B

_{y}, B

_{z}) = (A

_{x}

**i**A

_{y}

**j**A

_{z}

**k**).(B

_{x}

**i**B

_{y}

**j**B

_{z}

**k**)

=A_{x}**i**.B_{x}**i** A_{y}**j**.B_{y}**j** A_{z}**k**.B_{z}**k** 0

= A_{x}B_{x} A_{y}B_{y} A_{z}B_{z}

## No comments:

Post a Comment