Math 2433
Instructor: Ajit Bhand
Quiz 7A
Name:
December 3, 2009
Instructions
• Explain your answers clearly. Credit will be given only if you provide sufficient justification
for your answers.
• Non-graphing calculators are permitted.
• All phones, including iPhones and other smartphones must be turned off.
1. [4 points] (a) Describe a method for determining whether three points P, Q and R lie
on the same straight line.
(b) Describe a method for determining whether four points P, Q, R and S lie in the
same plane.
Solution:
(a) There are many ways to check this. If the vectors P Q, P R and QR are parallel,
the points lie on the same lie. It can also be checked if |P Q| + |QR| = |P R|.
(b) The points lie in the same plane if the scalar triple product of the vctors P Q, P R
and P R is zero.
2. [6 points] Determine whether the lines L1 and L2 are parallel, skew or intersecting.
y+1
z−3
x−2
=
=
3
1
2
x−1
y−2
z−5
L2 :
=
=
.
2
−1
−3
L1 :
Solution: The parametric equations for the lines are
L1 : x = 2 + 3t, y = −1 + t, z = 3 + 2t
L2 : x = 1 + 2s, 2 − s, z = 5 − 3s.
The vectors in the directions of L1 and L2 are < 3, 1, 2 > and < 2, −1, −3 > respectively.
Since they are not proportional, the lines are not parallel. To check if the lines are
interesecting, we equate the corresponding x and y components to get
2s − 3t = 1
s+t=3
from which we get s = 2 and t = 1. Substituting this in the z coordinates of each,
we see that they are different (3 + 2t = 5 6= 5 − 3s = −1). Hence, the lines do not
interesect. In conclusion, the given lines are skew.
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