vectors - What is the difference between dot and cross product?

1. 
   

   What is the difference between dot product and cross product?

   
   
   


2. 
   

   Why do we use cross product to find torque, why can't we use dot product?

   
   


3. 
   

   Also we use dot product to find work done and not cross product?

   
   

Answer

If we have a force $\vec{F}$ which acts upon a test particle along a curve $C$, then the work done is in the general case a line integral, given by,

$$W=\int_{C} \vec{F} \cdot \mathrm{d}\vec{r}$$

We may think of the integral as a summation over the contributions of the force along an infinitesimally small line element. The work done $W$ is a scalar quantity, and employing a cross product would not be sensible. On the other hand, torque is usually described as a vector, given by,

$$T = \vec{r} \times \vec{F}$$

If we employed the dot product, we would retrieve a scalar rather than a vector. In addition, recall torque is a description for the tendency for a system to rotate; a scalar quantity cannot fully capture this.