What is the formula for finding the dot product of two vectors?

The dot product of two vectors is calculated by multiplying the corresponding components of the two vectors and then summing those products. For two vectors ( \mathbf{u} ) and ( \mathbf{v} ) represented as ( \mathbf{u} = (u_1, u_2, u_3) ) and ( \mathbf{v} = (v_1, v_2, v_3) ), the dot product is given by the formula:

[ \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 ]

This formula reflects how the dot product measures not only the similarity in direction between the vectors but also scales it by their magnitudes. Each component's multiplication represents projection, and summing them accounts for contributions in all dimensions.

While other options present various mathematical concepts, they do not describe the dot product as clearly. For example, the second option involves the sine function and is related to finding the magnitude of the cross product, not the dot product. The third option simply adds the components without multiplying them, disregarding the necessary relationship for calculating the dot product. The fourth