Need help with finding an equation of a plane..?

Question

Here's the question..

Find an equation of the plane that passes through the line of intersection of the planes x - z = 1 & y + 2z = 3 and is perpendicular to the plane x + y - 2z = 1.

I hope someone could guide me along - I'm so confused!!

Answer 1

Sorry you are so confused. Let's take it step by step.
Please try to follow along and work through the problem with me.
Let the plane you want be ax + by + cz = d.
Our job will be to find a,b,c in terms of d, which
will then divide out.
Step 1.
The plane you want is perpendicular to x + y - 2z = 1.
That means that the normal vectors to both planes
are perpendicular. That means the dot product
of the normal vectors is 0, so we get
a + b - 2c = 0.

Step 2. Let's find the line of intersection, L, of the planes
x - z =1 and y + 2z = 3.
a). Find a point common to both planes. This
is easy. (2,1,1) will do nicely.
b). Now find a direction vector for the line.
It must be parallel to both planes. That
means it must be perpendicular to both
their normal vectors. How do we find such
a vector? Take the cross product of the
normal vectors of both planes.
We get

i j k
1 0 -1
0 1 2

= i -2j + k.
So an equation for L is
x = 2 + t
y = 1 - 2t
z = 1 + t

Step 3. L lies in our plane and 2 points
determine L, so to get 2 more conditions
for the plane, plug in 2 points of L.
If we choose t = 0 and t =1,
the final system of equations for our
plane is
2a + b + c = d
3a - b + 2c = d
a + b - 2c = 0

Adding the second and third of these equations
gives
4a = d,
a = d/4
Now add the first 2.
5a + 3c = 2d,
c = d/4.
Finally, we find b = d/4 also.
Plugging these into the equation of
our plane, dividing out d and multiplying by 4
we finally get
x + y + z = 4
as our final answer.
I agree, not the easiest problem to tackle!
Hope that helped.

Answer 2

it think you go the question work you may have missed typed
cause the last equation does not make sentence ti ti a polynomail and the other ones are binimials

from the last part a perpendicular line means the same slope but the y intercept is the reciprocal eg( if y intercept =2 then perpendicular line y intercept = - 1/2 )