MATH 2203 โQuiz 2
January 25, 2006
NAME____________________________________
Find an equation for the plane that contains the point P0(๎9;0;2) and
also contains the line, L, with parametric equations
x= 2 ๎3t
y= 5
z= 2 ๎t.
After you have found an equation for this plane, show how you check to see
that your answer is correct.
Solution: By setting t= 0 and t= 1, we see that the points P1(2;5;2)
and P2(๎1;5;1) are on the line L. Since ๎๎!
P0P1=h11;5;0iand ๎๎!
P0P2=
h8;5;๎1i, a vector that is perpendicular to the plane in question is
n=๎๎!
P0P1๎๎๎!
P0P2=๎
๎
๎
๎
๎
๎
i j k
11 5 0
8 5 ๎1
๎
๎
๎
๎
๎
๎
=๎5i+ 11j+ 15k.
If P(x; y; z)is any point in the plane in question, then n๎๎๎!
P0P= 0. Thus,
an equation for this plane is
h๎5;11;15i๎hx+ 9; y ; z ๎2i= 0
or
๎5 (x+ 9) + 11y+ 15 (z๎2) = 0
or
๎5x๎45 + 11y+ 15z๎30 = 0
or
๎5x+ 11y+ 15z= 75.
Let us check to see that our answer is correct:
๎5 (๎9) + 11 (0) + 15 (2) = 75
shows that the point P0is in this plane.
๎5 (2 ๎3t) + 11 (5) + 15 (2 ๎t) = ๎10 + 15t+ 55 + 30 ๎15t= 75
shows that the line Lis contained in this plane.
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