Assignment For Section 1.8

You should read the section before attempting any reading questions or problems. Remember that you are responsible for all definitions and examples. Also remember to keep the answers to the reading questions on a separate piece of paper; these will be randomly collected from time to time.

1)   Prove definition 1.8.1 makes sense by answering the following.

a)    Show that points satisfying  (x - x0).n = 0  form a plane and that x is in this plane.  Recall the definition of a plane from page 11.

b)    Show that n  is perpendicular to this plane (ie  show that for any vector v  in this plane,  n  is perpendicular to v).

2)   What is definition 1.8.1 analogous to in , that is if the vectors  n  and   x0  are in ,  what is
{ | (x - x0).n = 0 }?

3)   Consider definition 1.8.2.  Prove this is the line through   x0which is parallel to  m  by answering the following.

a)    Show that  x0  is a member of the defined sets of points: { | x = t m +  x0,  t }
b)    Show that the line formed by the above set of points is parallel to  m (ie  show that for any vector v  lying in this line,  m  is parallel to v).

4)   Let  a  and  b  be two linearly independent vectors and let  x0 be a given point.  Consider equation (1.8.9):

x = x0 + s a + t b    with  s,t .
Prove that x0 lies in the set of points defined by (1.8.9) and that this set is perpendicular to  a x b (see 1b).

Problems:

page 75     1, 2, 4, 8, 9, 11, 17, 23, 24, 29