CHAPTER 16-Dot Product

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.