Dot product (or scalar product) is a type of vector multiplication and dot product of two vectors (\(\vec { A }\) and \(\vec { B }\)), whose x-, y-, and z-components are known, can be directly calculated with following dot product formula. The result of a dot product is scalar and it has no longer a direction.

\(\vec { A } \cdot \vec { B }={ A}_{ x }{ B }_{ x}+{ A}_{ y }{ B }_{y}+{ A}_{ z }{ B }_{z}\)
From above formula, we see that the dot product of two vectors is the sum of the products of their respective components.

There is another way to find the dot product. If the vectors are in magnitues and the angle between the vectors form, the dot product formula is

\(\vec { A } \cdot \vec { B } =\left| A \right| \left| B \right| \cos { \theta }\).

Dot Product Formula

Vector \(\vec { B }\) is projected on vector \(\vec { A }\) and scalar product is calculated. The two methods give exactly same result. Cos θ and Cos (360-θ) are same so there is no difference to take one of these angles between A and B. It’s immediately seen from the formula that the result  can be larger than 0, smaller than 0 or equal to 0. If the two vectors are perpendicular (θ=π/2), then the dot product is zero.

Dot Product Examples

Example-1:  What is the scalar product of \(\vec { A }\) and \(\vec { B }\) ?

\(\vec { A } =3\hat { x } -5\hat { y } +6\hat { z }\)
\(\vec { B } =2\hat { y }\)
\(\vec { A } \cdot \vec { B } =3\ast 0+(-5)\ast 2+6\ast 0=-10\)


Example-2:  Find the dot product of \(\vec { A }\) and \(\vec { B }\).

\(\vec { A } =\hat { y }\) (Unit vector in the y direction)

\(\vec { B } =\hat { z }\) (Unit vector in the z direction)

\(\vec { A } \cdot \vec { B } =0\ast 0+1\ast 0+0\ast 1=0\)


Additional Resources