Fall 2002

Partial review of chapter 8:

Some key concepts from Sections 8.2, 8.3, and 8.4

*The parametric form of the line through two given points.*- Let
*A = (a,b)*and*B = (u,v)*be two points in the plane.- The
of the line joining*direction indicator**A*and*B*is*D = B - A.* - In terms of the coordinates,
*D = (u,v) - (a,b) = (u-a, v-b).*

(See Figure 8.3 in the text.)

- So, if a person starts travelling in a straight line from
*A,*and covers the distance from*A*to*B*in 1 unit of time, then the person's position at time*t*is given by the formula

*P(t) = A + tD,*or equivalently*P(t) = A + t(B - A).*We also refer to this formula as the

*parametric form of the line.*

- A line given parametrically with the same starting point
*A*and direction indicator

2*D*= (2*(c-a),*2*(d-b)),*then it is the same line, but the traveler is moving twice as fast.

(Also see, a recommended review problem).*Exercise 8.2.17*

- Two lines with the same direction indicator are parallel (or the same …).

Similarly, a line with direction indicator*kD,*where*k*is a real number, is parallel to a line with direction indicator*D*(unless it is the same line).

- Find the
and the*direction indicator*of each of the following lines:*parametric form**a.**(a,b)*and*(c,d).*

- The

- Let
*Vector addition, vector subtraction, and plane geometry.*

(Related toa recommended review problem.)*Exercise 8.2.24,*- Let
*A = (a,b)*and*B = (c,d)*be two points in the plane. - Then
*A+B = (a+c, b+d).*This is shown in the sketch below, for*A*= (2-1) and*B*= (3,2).

- What are the direction indicators of the following lines? (Try it for the particular points

*A*= (2-1) and*B*= (3,2) and also in general,*i.e.,*for abstract points*(a,b)*and*(c,d)*.) - The line joining (0,0) and
*A.* - The line joining
*B*and*A + B.* - The line joining (0,0) and
*B.* - The line joining
*A*and*A + B.*

- Which of the above pairs of lines are parallel? What kind of quadrilateral

is formed by the four points (0,0),*A, B,*and*A + B*?

- Let
*Finding the slope of a line from its parametric form.*

(Also see Exercise 8.2.14 anda recommended review problem.)*Exercise 8.2.19,*- Let a line be given parametrically, with direction indicator
*D = (c,d).* - If the starting point is
*A = (a,b),*then the position at time*t*is:

*P(t) = A + tD.* - For practice, sketch this with
*A*= (1,3) and*D*= (2, -1).

Calculate P(0) and P(1) in this case. - Use the answer from the previous part to calculate the slope of the line.
- Calculate P(0) and P(1) for
*P(t) = A + tD,*where*A = (a,b)*and*D = (c,d).* - Use the answer from the previous part to calculate the slope of the line.

(Simplify your answer as much as possible.)

- Let a line be given parametrically, with direction indicator
*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
. above to find the parametric form.*I*

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

- If the equation of a line is given as
*Norms and distances*- Let
*P = (x,y)*be a vector. - By definition, the
*norm*of*P*is . - If we think of
*P = (x,y)*as representing a point of the plane, then the Pythagorean theorem tells us that the norm of*P*is the distance from (0,0) to*P.* - If we think of a vector
*D*as being an arrow, then the norm ||*D*|| is just its*length.* - So, in general, we need
of numbers to specify a vector, but its norm is*an ordered pair**just a number.* - If
*P = (a,b)*and*Q = (c,d)*are two points, then ||*Q - P*||, the norm of the difference, is the distance from*P*to*Q:* *Q - P = (c-a, d-b),*so that:- For practice, sketch this with
*P*= (1,3) and*Q*= (2, -1), and calculate ||*Q - P*||.

- Let
*Segments*- Let the line joining
*A*and*B*be given in parametric form:

*P(t) = A + t(B - A).* - P(0) = A, and P(1) = B. Points of the
**segment***AB*are points*P(t),*where*t*ranges

through the numbers between 0 and 1. So,*t*= 0 and*t*= 1 give the endpoints, and is the midpoint. - Calculate for the following cases:
*A*= (-3,2) and*B*= (4,0)- For two "abstract" points
*A*and*B.*Simplify so that*A*and*B*each appear just once. (If you're using coordinates, get your answer into a form where each coordinate appears just once.) - Calculate for the following cases:
*A*= (-3,2) and*B*= (4,0)- For two "abstract" points
*A*and*B*. (Simplify your answer, as above).

- Let the line joining
*The dot product and orthogonality*

*Calculating the dot product*- If
*P = (a,b)*and*Q = (u,v),*then the*dot product**P*is given by the formula:**·**Q

*P*.**·**Q = au + bv

If*Sample:**P*= (1,3), and*Q*= (5,-3), then*P*= 1·5 + 3·(-3) = 5 - 9 = -4.**·**Q

So, the dot product of two vectors is just a number.

- Calculate
*P*in the following cases:**·**Q*P*= (2,-1) and*Q*= (4,3)

Sketch these two points in the plane, and draw their "position vectors",

draw an arrow pointing from the origin to each point.*i.e.,*

*P*= (2,4) and*Q=*(6,-3).

Sketch these two points in the plane, and draw their "position vectors",

draw an arrow pointing from the origin to each point.*i.e.,*

about the angle formed by the line joining (0,0) to*What can you observe**P*and the line joining (0,0) to*Q*?

*P*= (4,3) and*Q = P*= (4,3).

Also, calculate ||*P*|| and compare this with the dot product*P*.**·**P

*P*= (*a,b*) and*Q = P*= (*a,b*).

Also, calculate ||*P*|| and compare this with the dot product*P*.**·**P

- If
*Norm of a sum*- The solution of Exercise 8.4.4 says that:

||*P + Q*||^{2}= ||*P*||^{2}+ ||*Q*||^{2}+ 2*P***·**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*areIn other words:*orthogonal.*

*P*= 0 if and only if**·**Q*P*and*Q*are orthogonal.

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

- The solution of Exercise 8.4.4 says that:

Back to the class homepage