Question

Find a plane containing the point (5,4,-6) and the line of intersection of the planes -x+2y+8z=-57 and 5x-4y+7z=-27

I found the cross product of the two equations and got 46i+47j-6k.
I made x=0 and found the point (0, 615/46, -87/23).

The answer is -54x+36y-132z=666 but I do not know how to get there.

Answer 1

well, (0, 615/46, -87/23) isn't on the line

Answer 2

first pick two points that are on the intersection of the two planes

Answer 3

(-1,-5,-6) is one of them

Answer 4

So make one of the variables 0 and solve the equations?

Answer 5

you could, but you'll have fractions which make calculator more tedious

Answer 6

the other point is (45,42,-12)

Answer 7

How did you find those points?

Answer 8

I cheated XD. Wolfram alpha haha

Answer 9

haha. Okay.

Answer 10

It will take number theory to find those points, but for now, cheating is good enough XD

Answer 11

Knowing those two points, how do I find the equation of the plane?

Answer 12

now find two vectors using those two points and the point given

Answer 13

and then cross them

Answer 14

Okay. Let me try it.

Answer 15

got the two vectors? shouldn't take this long

Answer 16

I just crossed multiplied them.

Answer 17

crossed multiplied? you meant cross the two vectors?

Answer 18

Yes. And I got -54i+36j-132z

Answer 19

Anhoang: Consider carefully what info you need to define a plane.

Secondly, please re-write your sentence, "I found the cross product of the two equations and got 46i+47j-6k." One doesn't find the cross product of equations; one writes the cross product of two vectors. Mind showing us your work?

Now supposing that you correctly identify the two vectors whose cross product you're finding, and actually find their cross product. How will that cross product help you define the new plane you're seeking? The most common way to define a plane is
\[a(x-x _{0})+b(y-y _{0}+c(z-z _{0})=0,\]

where <a,b,c> is a vector NORMAL to the new plane and (x0,y0,z0) is a point IN that new plane. Have you actually found a vector NORMAL to the new plane?

Answer 20

good, your normal vector matches with the answer given. Now just write the equation for the plane and simplify them. You should get the same asnwer

Answer 21

I solved it by...
i j k
6 9 0
40 38 -6
(-54-0)i-(-36-0)j+(228-360)k = -54i+36y-132j

Answer 22

Thank you so much for your help!

Answer 23

yep :)

Answer 24

Congrats, anhoang! Good guidance, sourwing.