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

Answer 1

The intersection of 2 planes ( if there is) is a line, what is it? ( I mean the equation of that line?)

Answer 2

The question does not given any other information

Answer 3

sure, but we have a way to find it out, :) you are supposed to know it.

Answer 4

Would I need to find the normal vector? So the cross product of those two equations?

Answer 5

@sourwing please help

Answer 6

ok, he left. I don't have time, but let me try.

Answer 7

Thank you!

Answer 8

Since I tried solving the cross product and got 46i+47j-6k

Answer 9

let say P1 , P2 are plane 1, 2, so normal vector of P1 is n1 = < -1,2,8> and n2 = <5,-4,7> ok?

Answer 10

Okay

Answer 11

then n1 x n2 =?

Answer 12

I got 46i+47j-6k

Answer 13

Then do I need to make x=0 and solve for y and z?

Answer 14

That is the intersection line, and you need 2 points in that line, right?
but if you want to go on your way, you let x =0 for the equations of 2 planes, not this line
I mean -x+2y+8z=-57
and 5x -4y +7z = 27
let x =0 for both , solve for y and z

Answer 15

what do you have?

Answer 16

Hold on...

Answer 17

2y + 8z = -57
-4y +7z = 27
so, y = -615/46
z= -87/23

Answer 18

I am sorry, I don't have time, cannot wait for you :)

Answer 19

No worries!

Answer 20

so, you have a point (0, -615/46, -87/23)

Answer 21

And that would be my (x0,y0,z0) point?

Answer 22

And (46,47,-6) would be my (a,b,c) points?

Answer 23

therefore, the equation of the intersection line is
x = 46t
y =( -615/46) + 47t
z = (-87/23) -6t
now, just let t =0 to get point A (...,...,..)
and then t =1 to get point B (...,...,...)
accompanied with the given point (5,4,-6)
3points , so you can write down the equation of the required plane, right?

Answer 24

Got it. I'll try to go through that process then.

Answer 25

ok, good luck, sorry for not letting you work on your problem, I need my time to write my lab report. try to help because I started but sourwing didn't want to get over it. :)

Answer 26

Thank you so much for helping!

Answer 27

;)

Answer 28

I was in class at the time :DD