We have already seen the addition and subtraction of vectors. In this section, we’ll understand how we can define the product of two vectors.
Before formally defining the dot product, let us try to understand why (such a) product is required at all.
Consider a force acting on a block at an angle to the horizontal.
This block, as an effect of the force, is displaced through a horizontal distance . We can denote the displacement by the vector , which has a horizontal direction and has a magnitude .
In physics, you must have studied the concept of the work done by a force. This work done is the maximum if the force and the displacement caused by it are in the same direction, and zero if the force and the displacement are perpendicular. This suggests that we must consider the component of along to evaluate its work done.
From the figure above, the component of along the direction of is . The work done will then be
We denote the right hand side of by the dot product notation.
In general, for two vectors and inclined at an angle to each other, their dot product is defined as
It is very important to understand that is a scalar quantity
You can think of in this way: it is a measure of the “effect” of one vector along the other. For two vectors of fixed magnitude , their dot product will decrease in magnitude as increases from to (or decreases from to ).
We can write as
Thus, is the product of the modulus of either vector and the projection of the other in its direction.