Download MATH 311 - Vector Analysis BONUS # 1: The parametric equation of

MATH 311 - Vector Analysis
BONUS # 1:
The parametric equation of the plane Π that goes through the point P~0 = (x0 , y0 , z0 ) and which
~1 = (a1 , b1 , c1 ), and
is parallel to the plane defined by the non-colinear and non-zero vectors V
~
V2 = (a2 , b2 , c2 ) is given by the vector equation:
~ 1 + sV
~2 ,
P~ = P~0 + tV
s, t ∈ IR.
In components this vector equation becomes 3 equations:
x = x0 + ta1 + sa2
y = y0 + tb1 + sb2
z = z0 + tc1 + sc2
Find a non-parametric equation of the plane not using cross-product, but instead by eliminating
the parameters s, t in the above equations and reducing to just one equation that will involve the
variables (x, y, z), and the data (x0 , y0 , z0 ), (a1 , b1 , c1 ), and (a2 , b2 , c2 ). If you are careful in your
calculations you will obtain an equation of the form
A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0,
where the coefficient A depends only on b1 , b2 , c1 , c2 , the coefficient B depends only on a1 , a2 , c1 , c2 ,
and the coefficient C depends only on a1 , a2 , b1 , b2 .
~1 × V
~2 .
Show that the vector (A, B, C) is parallel to the cross-product V
This Bonus is due on Wednesday August 31st, 2005.
Please do it in a separate sheet from your usual homework since I will grade it, not the grader.