A vector is a quantity that has magnitude (size) and direction. For example, if a car is going 50 mph due north, the vector describing the velocity of the car is (50 mph, north). We note that we need two numbers for vectors. In three dimensions, we need three numbers. We can also give the two numbers as the magnitude in the x direction and the magnitude in the y direction. The velocity of the car discussed above is then (0 mph, 50 mph), as it is only in the y direction.

 

We know how to add vectors. Let A and B be two vectors. Vectors are written in boldface in print, and underlined A when we write using pen or chalk.

 

The sum of A and B is another vector C.  We can find C by saying that the x component of C is the sum of the x components of A and B. Easy. Using component notation, where Ai is the ith component of A, we say Ai + Bi = Ci.

 

How do we multiply vectors? How do we multiply directions?  Recall what mathematics is. Say whatever you want as long as you are consistent. Logically, we can either say that the product of two vectors is a vector or a number. Here is how we define them. We say that if the product is a number we call it the dot product: A · B, which is c (a number). The other logical possibility is that the product is a vector. In this case, we call it the cross product: A × B. This is a vector C.

 

If the two vectors are parallel, then we just multiply the magnitudes to get the dot product. If they are not parallel, we ignore the perpendicular components. Using component notation,

Ai  Bi = C, where we sum over the ith components.

 

If the vectors are perpendicular, then we again multiply the magnitudes to get the magnitude of the cross product. We ignore any parallel components. What is the direction of the vector C? What unique direction is there which is different from the directions of A and B? When I ask this in class, students say a line drawn at an angle. No! The unique vector is perpendicular to the board. If I draw A and B on the board, the direction of C is perpendicular to the board. I put a ruler on the board, and this is the direction of C. We can also write this in component notation, but will not do it here.

 

The next question is where does it point – out of the board or in the board? The answer is the right hand screw. If we open a bottle with a screw lid, we apply a torque T, which is defined as T = r × F, where r is the radius of the lid, and F is the force of my hand. If I turn to the right, the lid comes up. The direction of T is straight up.

 

Note that in order to open the bottle, we have to apply a force perpendicular to the radius. If we apply a force parallel to the radius, the bottle will move, but the lid will stay on.

 

Normally, we apply two forces, equal and opposite, so that the bottle does not move. The torques due to the two forces are in the same direction, and add. This we see from the definition.

 

Torque is like force, but it has to do with rotation, like turning the lid off the bottle. We can do the same mathematical thing with momentum, and define angular momentum L.

 

       L = r × T.