Ma 237
Fall 2010
Homework 1
Carter
1. Determine the solution set for each of the equations below under the following scenarios:
• we consider the solution set as a point on the line;
• we consider the solution set as a line in the plane;
• we consider the solution set as a plane in space;
• x = x1 is the first of n variables x1 , x2 , . . . , xn .
(a)
3x = 2
(b)
2x = 3
(c)
2x = 8
(d)
5x = 35
2. Express the solution set to the following equations as a point plus parameter times
direction where the parameter is (a) the free variable y; and (b) the free variable x.
Please do the problems in this order.
(a)
3x + 4y = 24
(b)
2x − 5y = 30
(c)
x+y =1
(d)
5x − 4y = 20
For each of these lines determine a vector that is perpendicular to the line. Furthermore, determine the intersection of the line with each of the coordinate axes: x = 0
(the y-axis) and y = 0 (the x-axis).
3. Express the solutions set to the following equations as a point in space plus the sum
of two parameters times directions as in the example below. Use y and z as your free
parameters.
(a)
3x + 4y + 8z = 24
(b)
2x − 5y − 6z = 30
(c)
x+y+z =1
(d)
5x + 4y − 2z = 20
1
Determine a vector that is perpendicular to the plane defined by the equation.
Example
2x + 3y + 5z = 30
3
5
y− z
2
2
0+ 1 y +0 z
0+ 0 y +1 z
x = 15 −
y =
z =
x
15
− 23
− 52








 y  =  0 +y 1 +z 0 
z
0
0
1








The vector [2, 3, 5] is perpendicular to the given plane since
− 32
3


[2, 3, 5] ·  1  = 2 × −
+3×1+5×0=0
2
0

and

− 52
5

[2, 3, 5] · 
+ 3 × 0 + 5 × 1 = 0.
 0 =2× −
2
1


For each of these planes, determine a vector that is perpendicular to the plane. Furthermore, determine the line of intersection between this plane and each of the coordinate
planes x = 0 (the y, z-plane), y = 0 (the x, z-plane), and z = 0 (the x, y-plane).
4. Mimic the example and exercises above to solve the equation
2x + 3y + 4z + 5w = 60
for x. Find a particular solution. Also determine three directions which “span” the
solid of solutions. Determine the intersection of this solid with each of the coordinate
solids: x = 0, y = 0, and z = 0. Can you determine a vector that is perpendicular to
this solid?
5. Solve the equation
a1 x1 + a2 x2 + · · · + an xn = b
for x1 . write your solution in a fashion that mimics the worked example. Give the
intersections between this plane and the coordinate planes xj = 0. Under what circumstances will there be no such intersection?
2