Find a set of parametric equations for the line of intersection of the planes.
6x-3y+z=5 and -x+y+5z=5[/B]
The cross product formula
The formula for the parametric equations of a line in three dimensional space:
x=x1+at, y=y1+bt, z=z1+ct
Knowing the fact that the coefficients in front of the variables in the equation of a plane are the components for the normal vector of the plane
The Attempt at a Solution
I found the cross product between the normal of the two plane. The resulting vector is going to function as the direction vector for the intersection line.
a= the normal vector of the first plane
b=the normal vector of the second plane
finding a point on the intersection line:
We could do this by forming a system of equations. We get rid of on of the variables by saying that at some point the value of z in both planes will be the same and at some point it will be zero. This will give us the system of equations:
Would solve for one of the variable by using the method of elimination. We could multiply the second equation by 6. This will give us:
We could solve for y:
we could substitute the value of y into the second equation and that will give us a value for x:
Now, we have the point ((20/3), (35/3), 0)
From this point it is really easy to plug the numbers into the parametric equations of a line. I referred to the solution of this problem. We both had the same values for the direction vector, but the point on the line was different. This makes the parametric equations of the intersection line have different values for x1, y1, z1. I was wondering if there is a unique answer to this problem? My thought is that there isn't, because we could choose any point on the line and use it to write the parametric equations of the intersection line. Am at right on this?