Question

Grade 12 math (vectors): intersection of two planes?

Determine the Cartesian equation of the the plane that is parallel to the line with equation x = - 2y = 3z, and that contains the line of intersection of the planes with equations x - y + z =1 and 2y - z =0.

answer: 8x+14y -3z -8=0

How do you solve this question? Can someone please help me?

Answer 1

ok first I guess you need to find the line of intersection of the 2 planes

Answer 2

the normal vectors of the planes are (1, -1, 1) and (0, 2, -1)

Answer 3

yell if something is not clear or not correct

Answer 4

the cross product of these vectors will be perpendicular to both vectors

Answer 5

do you know how to get the cross product?

Answer 6

yes I know the cross product

Answer 7

ok cool, so it will be: (-1, -1, 2)

Answer 8

this is a direction vector for the line

Answer 9

now we need to find a point of intersection:
x - y + z =1 and 2y - z =0.

Answer 10

y=1, z=2, x=0

Answer 11

that is a good one for us

Answer 12

so the equation of the line is: (0,1,2) + t(-1, -1, 2)

Answer 13

hmm I dont like this

Answer 14

yup i'm following so far

Answer 15

I see the problem, the cross product is (-1 , 1, 2)
so the equation of the line is (0,1,2) + t(-1,1, 2)

Answer 16

that is good now

Answer 17

so that's the end?

Answer 18

no no, we only have the line of intersection now

Answer 19

I am not that good with this but I try :-)

Answer 20

oh it's alright, at least we got more than half of it done :)

Answer 21

sorry, brb for 30 min

Answer 22

now I am bit stuck

Answer 23

if the two planes are parallel than there normal vector is the same

Answer 24

I dont know, sorry... :(

Answer 25

that's alright :) thanks anyways andras!