- #1
space-time
- 218
- 4
Homework Statement
Find an equation of the plane that passes through the line of intersection of the two planes 3x + 7y + 2z= 0 and -9x - 8y - 6z = 9, and is perpendicular to the plane -2x - 6y + 3z = -10.
Homework Equations
The Attempt at a Solution
Here is what I did:
I first took the two normal vectors (n1 and n2) of the first two planes mentioned.
n1 = (3, 7, 2)
n2 = (-9, -8, -6)
I then took the cross product of these two vectors:
n1 × n2 = (-26, 0, 39) We can call this vector V.
I then found the normal vector of the plane that is mentioned in the end of the problem (which I will denote as n3)
n3 = (-2, -6, 3)
Next I took the cross product between V and n3.
V × n3 = (234, 0, 156)
so then my normal vector for the plane that will be my solution is:
nfinal = (234, 0, 156)
I then found a point on the line of intersection in the plane by first setting z=0 and then solving a system of equations for x and y. Here is the system:
3x + 7y = 0
-9x - 8y = 9
solving for x and y yields:
x = -189/117 , y= 9/13 (and of course z = 0)
Now that I have my point and my direction vector, I can find the equation of the plane. I got:
(234, 0, 156) ⋅ ( x + 189/117 , y - 9/13 , z) = 0
which expands to:
(234x +378) + (0 - 0) + 156z = 0
which simplifies to
234x + 378 + 156z =0
or simply
234x + 156z = -378
That was the final answer that I plugged into the software for the equation of the desired plane. There were two answers that the problem wanted: They wanted the normal vector of the desired plane and the equation of the desired plane.
When I plugged in my normal vector of (234, 0, 156) and my equation 234x + 156z = -378, the software said that I got the normal vector right, but got the equation wrong.
Why?! I don't know what I did wrong here. Furthermore, I worked another example of this type of problem that simply had different numbers and equations with a friend earlier, and I got it right using the exact same process I used here! I've also looked up other examples of this online and every other example used the exact same process!
Can someone please help me with this (because I am really ticked off and don't know what is wrong)?