Download Math 341 Geometric problems 1. Find a cubic polynomial whose

Math 341
Geometric problems
1. Find a cubic polynomial whose graph passes through the points (−1, 4), (1, 2), and (2, 0). (Note: there
are infinitely many such cubics—you are only required to give one of them).
2. (a) Write down a parametric equation for the line l in R3 which passes through (1, 2, −4) and (3, 1, 5).
(b) Determine if the point (−3, 4, −20) is on the above line.
(c) The point Q = (1, 1, 3) is not on the line. Write down a parametric equation for a line with passes
through Q and is parallel to l.
3. Write down a parametric equation for the line y = 2x − 7 in R2 .
4. Here are two parametric equations for lines in R3 :
   
 
x
1
5
y  = −3 + t  3 
and
z
2
−3
   
 
x
−8
2
y  =  2  + t  7  .
z
9
−1
Determine if these two lines intersect, and if so find the point of intersection.
5. (a) Write down a parametric equation for the plane in R3 containing the points (3, 3, 1), (4, −1, 2), and
(0, −1, 3).
(b) Write down a normal equation for this same plane.
(c) Write down a parametric equation for a line in R3 passing through the point (1, 1, 2) which does
not intersect the plane in (a).
(d) Write down a parametric equation for a line in R3 passing through the point (1, 1, 2) and perpendicular to the plane in (a).
6. (a) Write down a parametric equation for the plane in R3 defined by the equation 3x − 5y + 2z = 4.
(b) Write down a parametric equation for a line in R3 passing through the point (1, 1, 2) and perpendicular to the plane in (b).
7. Determine if the plane 2x − y − 3z = 4 intersects the line
   
 
3
x
10
y  =  6  + t 1 .
11
4
z
If they do intersect, find the point of intersection.
8. Consider the plane and the line defined by the equations
   
 
 
x
1
1
1
y  = 1 + s 0 + t 3
and
z
3
1
4
   
 
x
2
3
y  = 1 + t 3 .
z
5
6
Determine if the plane intersects the line, and if so find the point of intersection.
9. The planes 3x − y + 2z = 1 and 2x + y + z = 4 intersect in a line. Determine a parametric equation for
this line.
10. Repeat question #4 but for the lines in R5 given by
   
 
x
−2
2
y   1 
1
   
 
z  =  4  + t  1 
and
   
 
w  0 
−2
u
1
0
11. Decide if the line
   
 
x
0
−9
 y  1
−6
   
 
 z  = 6 + t −3 .
   
 
w 0
 12 
u
3
3
   
 
x
1
1
2
 y  −1
  
 
l:
 z  =  4  + s 1
w
6
1
intersects the plane
   
 
 
x
1
3
−1
 y  −1
0
1
  
 
 
M :
 z  =  0  + s 1 + t  2  ,
w
1
2
2
and if so find the point of intersection.
12. (a) Write down a parametric equation for the solution set of 3x − 2y + 4z = 1.
(b) Is (2, 1, 1) in this solution set? Explain how you know.
13. Write down a parametric equation for the solution set of 2x + y − 4z + 3w = 2.
14. Write down a parametric equation for the line 2x + 5y = 8 in R2 .
15. Find a parametric expression for the plane in R6 containing the three points P1 = (0, 5, 3, −1, 2, 1),
P2 = (3, −1, 2, 6, 3, 1), and P3 = (4, 3, −2, 0, 1, 1).
16. Consider the following lines in R4 :
 
   
3
−1
x
1
y   2 
 
  
L1 : 
z  =  1  + t  1 
−1
0
w
and
L2 :
 
   
5
3
x
4
 y  1
  =   + t .
4
 z  0
−3
0
w
Decide if L1 and L2 intersect, and if so determine the point of intersection.
17. Consider the plane in R5 given parametrically by
 
 
   
2
−3
1
x
1
1
y   0 
 
 
   
 
 
  
M: 
 z  =  3  + t −5 + s −3 .
6
1
w  2 
8
3
1
u
Is the point (1, 5, 3, 2, 1) on this plane?
18. Consider the two planes in R4 :
 
 
   
0
−3
1
x
1
1
y   2 
 
 
  
M1 : 
z  =  3  + t  4  + s  3 
−2
2
−4
w
and
M2 :
 
 
   
1
0
−7
x
1
6
 y  −3
  =   + t  + s .
0
10
z   4 
1
1
−5
w
These planes happen to intersect in a line: Find a parametric expression for this line.
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19. Consider the plane in R4 given as
N:
   
 
 
x
1
−3
5
 y  2
1
4
  =   + s  + t .
 z  1
7
1
w
3
1
1
The point P = (4, 7, 9, 5) is not on this plane. Find a parametric expression for a plane which passes
through P but which doesn’t meet the plane N . (Hint: There is a way to do this which involves almost
no work. Draw a picture, and think.)
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