The scalar product, commonly known as the dot product, is one of the fundamental ways to multiply two vectors. This operation takes two vectors and returns a single scalar number. The dot product has important physical interpretations, most notably in the calculation of mechanical work and the projection of one vector onto another.
The dot product of two vectors, and , is a scalar quantity equal to the product of their magnitudes and the cosine of the angle () between them.
Formula:
Where:
The dot product can be geometrically interpreted as the magnitude of one vector multiplied by the projection of the second vector onto the first.
Here, is the scalar projection of vector onto vector . This tells us how much of vector points in the same direction as vector .
If two vectors are expressed in terms of their rectangular components, their dot product can be calculated by multiplying their corresponding components and summing the results.
Rectangular Components Of A Vector→
Given:
The dot product is:
This formula is derived from the properties of the unit vectors:
| Property | Description | Mathematical Expression |
|---|---|---|
| Commutative | The order of the vectors does not matter. | |
| Parallel Vectors | If two vectors are parallel (), their dot product is the product of their magnitudes. | |
| Orthogonal Vectors | If two vectors are perpendicular (), their dot product is zero. | |
| Anti-Parallel Vectors | If two vectors are in opposite directions (), their dot product is the negative product of their magnitudes. | |
| Self-Dot Product | The dot product of a vector with itself gives the square of its magnitude. |
The dot product is the standard way to calculate the mechanical work done by a constant force.
This formula correctly captures that only the component of the force parallel to the displacement does work.
The dot product formula can be rearranged to find the angle between two vectors:
Power can be expressed as the dot product of force and velocity.