Equation of plane parallel vectors

[Jbreezy]

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?

 

[jing2178]

Looks good to me