## Math 3118, Section 1 Fall 2002 Partial review of chapter 8:       Some key concepts from Sections 8.2, 8.3, and 8.4

1. The parametric form of the line through two given points.
2. Vector addition, vector subtraction, and plane geometry.
(Related to Exercise 8.2.24, a recommended review problem.)
3. Finding the slope of a line from its parametric form.
(Also see Exercise 8.2.14 and Exercise 8.2.19, a recommended review problem.)

4. Finding the parametric form of a line from its slope-intercept equation.
(Also see Exercise 8.2.18)
• If the equation of a line is given as  y = mx + b, (this is called the  slope-intercept form of the equation) we can represent the line parametrically as follows:
• Find two points on the line (for instance by substituting two different x-values and then calculating the corresponding y-values).
• Use the methods of I. above to find the parametric form.

• So, the conclusion from III and IV is that it's possible to go back and forth between the two ways of describing a line.

5. Norms and distances
6. Segments
7. The dot product and orthogonality

• Calculating the dot product
• Norm of a sum
• The solution of Exercise 8.4.4 says that:
||P + Q||2 = ||P||2 + ||Q||2 + 2P·Q.
Thus, we need to add 2 times the dot product as a "correction term".

• If the line from  (0,0)  to  P is perpendicular to the line from  (0,0)  to  Q, then the parallelogram formed by   (0,0), P, Q, and  P+Q is actually a rectangle. So, th e Pythagorean theorem tells us that  ||P + Q||2 = ||P||2 + ||Q||2 in this case.
Thus, the correction term must be zero if these two lines are perpendicular. We express this by saying that  P and  Q are orthogonal. In other words:
P·Q= 0  if and only if  P and  Q are orthogonal.

• Is each pair orthogonal?  (compare with your sketches from above.)
1. P= (2,-1)  and  Q= (4,3)
2. P= (2,3)  and  Q= (3,-2).