Question

The plane through (1,2-1) that is perpendicular to the line of intersection of the planes 2x+y+z= 2 and x+2y+z = 3

Answer 1

Find the equation of the plane

Answer 2

The thing i don'y understand is that when you take cross product of the two normals to get the desired normal which is (-1,-1,3) why do we take it as (1,1,-3)?

Answer 3

it doesnt matter, they're direction ratios, so any of (-1k, -1k, 3k) will work

Answer 4

maybe you're changing it to(1, 1, -3) cuz it has fewer negatives

Answer 5

but the equation would be different then right?

Answer 6

Its done that way in the manual

Answer 7

how, can we walk through and see wat we get,
i dont remember exactly how to actually solve this

Answer 8

i remember :)

Answer 9

we get the two normals from the equations and then take their cross product. Then use that normal the point given to get the equation

Answer 10

the intersection of given planes is a line,
since this line is perpendicular to the required plane,
the equation wud be : a(x-x1)+b(y-y1)+c(z-z1) = 0

since we knw one point, we're done.

im thinking we can solve as above

Answer 11

a plane that is perp to a line, uses the lines direction vector as its normal .... yep

Answer 12

define 2 points to establish a vector between

Answer 13

we have a system of equations:

2x+y+z= 2
x+2y+z = 3

if we hold one variable constant, this reduces to a 2x2 system.

let x=0, solve for y,z
let z=0, solve for x,y

we then have 2 points to vector

Answer 14

yep thats the equation. we want (a,b,c) which we get through the cross product of (2,1,1) and (1,2,1)

Answer 15

or that .. yeah

Answer 16

but why take negative of the normal that we get

Answer 17

The cross product comes out as (-1,-1,3) but the normal is (1,1,-3)

Answer 18

-(z+2x+2y)
x y z x y
2 1 1 2 1
1 2 1 1 2
+(x+y+4z)


x +y+4z
-2x-2y -z
-----------
-1,-1,3

you do realize that we could have also gotten 1,1,-3 if we had swapped rows right?

Answer 19

by convention, a direction vector (a,b,c) is built such that a>=0, and a,b,c relative primes

Answer 20

please explain a little more

Answer 21

by convention ... as a rule of thumb that everyone agrees on ... becuase we like to have some sort of uniformity of results ... becuase this is what people have agreed to use ... etc,etc,etc.

a direction vector (a,b,c) is built such that a>=0, and a,b,c relative primes

Answer 22

All above show u same direction, (i missed putting arrows in other direction)

Answer 23

using any form of k(1,1,-3) will produce the same plane for any nonzero k

Answer 24

asking why we do it is the same as asking why everyone drives on the right side of the road in America .... it is simply something we all have agreed to do to make an otherwise chaotix system organized