1. Useful Math

2. dot product (or scalar product) of two vectors

12. 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.

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

14. 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

15. 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

16. 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.

20. 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 0.002. In practice this means that x can be regarded as effectively zero in comparison with R.