Useful Math

dot product (or scalar product) of two vectors

A photographer wants to get a picture of the upper three stories of a four- story apartment building. Each story is 15 feet tall (4.57m). She stands 20m away and holds her camera at a height of 1.1m above the ground.

What must be the minimum angular field of view of the camera if she is to get the picture?

A photographer wants to get a picture of the upper three stories of a four- story apartment building. Each story is 15 feet tall (4.57m). She stands 20m away and holds her camera at a height of 1.1m above the ground. What must be the minimum angular field of view of the camera if she is to get the picture? y x

solution: Draw two vectors from the camera to the building. Using the coordinate system shown, a vector to the top of the first story is and a vector to the top of the building is The angle between these vectors is

Binomial Formula Very often in physics we have to evaluate quantities of the form (1) (a +b) p where b is small compared to a (this is written as b << a and in practice means that b is smaller than b < a /10 ). The quantity in (1) is then very close to a p, but we often need a better approximation. Factor out a to get (a +b) p = a p (1 +  ) p where  = b/a <<1. Then perform a Taylor series expansion about  = 0.

This says that if d is small compared to R, x is MUCH smaller compared to R. For example, a typical glass lens might have R = 10cm, d = 1cm, so d/R = 0.1 Then x/R would be smaller than In practice this means that x can be regarded as effectively zero in comparison with R.