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A solution to finding the equation of a plane that contains a given point (p0) and a line with parametric equations. How to find the normal vector of the plane and derive the equation using vector algebra. It also checks the correctness of the answer by verifying that the point and the line are indeed in the plane.
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MATH 2203 รฑQuiz 2 January 25, 2006
NAME____________________________________ Find an equation for the plane that contains the point P 0 ( 9 ; 0 ; 2) and also contains the line, L, with parametric equations
x = 2 3 t 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 P 1 (2; 5 ; 2)
and P 2 ( 1 ; 5 ; 1) are on the line L. Since
P 0 P 1 = h 11 ; 5 ; 0 i and
h 8 ; 5 ; 1 i, a vector that is perpendicular to the plane in question is
n =
i j k 11 5 0 8 5 1
= 5 i + 11j + 15k.
If P (x; y; z) is any point in the plane in question, then n
P 0 P = 0. Thus, an equation for this plane is
h 5 ; 11 ; 15 i hx + 9; y; z 2 i = 0
or 5 (x + 9) + 11y + 15 (z 2) = 0
or 5 x 45 + 11y + 15z 30 = 0
or 5 x + 11y + 15z = 75.
Let us check to see that our answer is correct: