Calculate Product of Three Dimensional Vectors

The two vector A and B for dot or scalar product.

The dot product is a form of multiplication that involves two vectors having the same number of components. To determine the dot product of two vectors, we always multiply like components, and find their sum.

Example :

Let consider value for vector A as (2 , 3 , 4) and B as (4 , 6 , 5)

The A*B = a₁* b₁ + a₂* b₂ + a₃* b₃

= 2x4 + 3x6 + 4x5

= 8 + 18 + 20

= 46

Formula

The 3D vectors are using the x-y-z axes.

α is the angle between u and the x-axis.

β is the angle between u and the y-axis.

γ is the angle between u and the z-axis.

Therefore scalar product for unit vector:

u1 = u ⋅ i = 1 × 1 × cos α = cos α

u2 = u⋅ j = 1 × 1 × cos β = cos β

u3 = u ⋅ k = 1 × 1 × cos γ = cos γ

the unit vector u can be written as:

u = cos α i + cos β j + cos γ k