The Analytic or Component Method

Suppose we have two given vectors = (A_x, A_y) and = (B_x, B_y), as well as a simple number c. The vector notation is supposed to indicate a pair of cartesian coordinates. We can then define rules for calculating with vectors:

1. Addition of two vectors:
   If the vector = (C_x, C_y) is the sum of the two vectors and , = + ,

   then its components are given by:

   C_x = A_x + B_x,
   C_y = A_y + B_y,

   that is to say that one can simply obtain the x- and y-components of the sum vector as the sum of the components of the individual vectors.

2. Subtraction of two vectors:
   If the vector = (D_x, D_y) is the difference of the two vectors and , = - ,

   then its components are given by:

   D_x = A_x - B_x,
   D_y = A_y - B_y.

   This is basically identical to the addition procedure, but of course here the order is important.

3. Multiplication with a constant:
   If the vector = (E_x, E_y) is c times the vector , = c $\cdot$ ,

   then its components are given by:

   E_x = c $\cdot$ A_x,
   E_y = c $\cdot$ A_y.

   For the multiplication, you thus need to multiply each component of the vector with the same number.
   
   It is easy to show that in this case, the length of the vector is also c times the length of vector :

   E = c $\cdot$ A

Remember that if you know the x- and y-components of a vector , you know its length A and the angle q with respect to the x-axis.

The proof of all of these statements is pretty straightforward, but we omit it here. This is a physics class, not a mathematics class!

© MultiMedia Physics, 1999