2.15 : Dot Product

The dot product is an essential concept in mathematics and physics.

In engineering, the dot product of any two vectors is the product of the magnitudes of the vectors and the cosine of the angle between them. It is denoted by a dot symbol between the two vectors.

Consider a vehicle pulling an object along the ground using a rope. If the rope makes an angle with the horizontal axis, the work done can be calculated using the dot product of the force applied and the object's displacement.

The dot product of two vectors expressed in Cartesian form can be determined by multiplying their corresponding x, y, and z components and adding their products algebraically. This is given by the formula for the dot product of vectors A and B: 

If the angle between two vectors is unknown, the dot product can help determine the angle using the inverse cosine function. Suppose A and B are two vectors, and the angle between them can be determined by:

The dot product follows the commutative and distributive laws of addition and multiplication. The dot product is a fundamental operation used in many areas of physics and engineering. It can calculate work done, find the angle between vectors, and determine force components along a specific direction. With its many applications, the dot product is an important concept in vector algebra.