- #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 (n

_{1}and n

_{2}) of the first two planes mentioned.

n

_{1}= (3, 7, 2)

n

_{2}= (-9, -8, -6)

I then took the cross product of these two vectors:

n

_{1}× n

_{2}= (-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 n

_{3})

n

_{3}= (-2, -6, 3)

Next I took the cross product between V and n

_{3}.

V × n

_{3}= (234, 0, 156)

so then my normal vector for the plane that will be my solution is:

n

_{final}= (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)?