Question

Find an equation of the plane containing points R(1,2,3), T(-3,2,3), M(2,0,5).

So I got 0x+y+z=5

Then, find the distance from the place in #9 to point S(-1,2,5). Where do I go from here?

Answer 1

how did you obtain the equation of the plane

Answer 2

found the vector RT = <-4,0,0> and vector RM = <1,-2,2> then did the cross product to get 0i+8j+8k. Then I did vector RM (dot) <0,8,8> which is how I got 0x+1y+1z=5

Answer 3

was there a simpler way to do it?

Answer 4

yeah if you were give three points in (x1,y1,z1) (x2,y2,z2) (x3,y3,z3)
the determinant with first row x-x1 y-y1 z-z1
second row x2-x1 y2-y1 z2-z1
third row x3-x1 y3-y1 z3-z1

will give you the equation of the plane

and the distance between a plane and a point D=(|ax0 +by0+cz0+d| )/sqrt(a^2+b^2+c^2)

where a b c are the direction cosines of the plane in the normal form and x0 y0 z0 are coordinates of the point

Answer 5

i still dont understand

sorry :(

Answer 6

that will give you the equation of the plane passing through three points (x1,y1,z1) (x2,y2,z2) (x3,y3,z3)

Answer 7

\[|ax _{0}+by_{0}+cz_{0}+d| / \sqrt{a^2 +b^2+c^2}\]

Answer 8

is the equation that will give you the shortest distance between a point (x0,y0,z0) and a plane ax+by+cz+d=0

Answer 9

ok using your diagram I got x-1, y-2, z-3, etc

but I am confused as to how to use this in the equation ax_0?

thank you for your help!!!

Answer 10

the diagram is to find the equation of the plane which you already did
i just stated an easier way to do it

now you have a point S(-1,2,5) and a plane 0x+y+z-5=0
here x0 = -1
y0 = 2
z0 = 5

and a, b,c,d are 0,1,1,-5 respectively
substitute them in the equation of shortest distance between a plane and a point and you will get the answer

Answer 11

btw you know how to evaluate a determinant right?

Answer 12

no

Answer 13

do you know what a matrix is?

Answer 14

yes

Answer 15

on substituting the values this is how it'll look
(0)(-1) +(1)(2) +(1)(5) -5 /square root of (0)^2+(1)^2+(1)^2

Answer 16

ohh ok. thank you :)

Answer 17

did you get it

Answer 18

yes i got 2/root(2)

Answer 19

yep you got it

Answer 20

thanks!!

Answer 21

@Aenn88 , for future reference

http://en.wikipedia.org/wiki/Plane_(geometry)#Defining_a_plane_through_three_points

Answer 22

thanks for the link! thats not anywhere in my book. so much more simple

Answer 23

:)