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Jbreezy
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Homework Statement
The vectors a= <-4,3,3> and b = <2,-6,-5> are parallel to a plane PI and R is a point on
with position vector <104,8,-6> . Find the Cartesian equation of the plane. What is the
distance of the plane from the origin?
Homework Equations
The Attempt at a Solution
This is my thinking.
Since we are told the vectors a and b are parallel to the plane if we cross them we can get a vector perpendicular to the plane. We can turn that into a unit vector and dot it with the point r.
So,
a cross b = <3,-14,18> : call this vector v
v = √(3^2) + (-14^2)+(18)^2 = 23
So v(hat) = 1/23<3,-14,18>
Then to get the equation of the plane we can do
x dot v(hat) = r dot v(hat)
<x,y,z> dot 1/23<3,-14,18> = <104,8,-6> dot <1/23<3,-14,18>
So I got
3/23x - 14/23y + 18/23z = 92/23
Then just multiply through to clear out the fraction
3x - 14y +18z = 92
and the distance from the plane to the origin is 4 units because r dot v hat is 4
Did I do this correct?