Show that the points (-1,4,-3) , (3,2,-5) , (-3,8,-5) and (-3,2,1) are coplanar?

2 Answers
Feb 27, 2018

They are not coplanar . Still i could always be wrong !

Explanation:

using the four points , we can create 3 vectors with one of the point as common origin.

A(-1,4,-3) ; B(3,2,-5); C(-3,8,-5); D(-3,2,1)

i take A as common origin (doesn't matter which one you take) .
So i create 3 vectors AB , AC , and AD.
vec (AB) = <3-(-1),2-4,-5-(-3)> = <4,-2,-2>
vec(AC) = <-3-(-1),8-4,-5-(-3)> = <-2 , 4, -2>
vec(AD) = <-3-(-1),2-4,1-(-3)> = <4,-2,4>

Take take the cross product of any of the two vectors(this will give you a resulting vector which is normal to the plane ) and perform the dot product of the result with the remaining vector .

If you get 0 as the result , then the four points are coplanar

I'm taking cross product of AB and AC
vec(AB) X vec(AC) = <-12,-4,12> . I have skipped the calculations.

now dot product this result with AD
vec(res) . vec(AD) = <-12,-4,12> . <4,-2,4>
= 8 cancel= 0.
Therefore they are not coplanar .
I can always be wrong !

Feb 27, 2018

" "A-=(-1,+4,-3)
" "B-=(+3,+2,-5)
" "C-=(-3,+8,-5)
" "D-=(-3,+2,+1)
lie in a single plane

Explanation:

Let
" "A-=(-1,+4,-3)
" "B-=(+3,+2,-5)
" "vec(AB)=(3-(-1))hati+(2-4)hatj+(-5-(-3))hatk
vec(AB)=4hati-2hatj-2hatk

" "A-=(-1,+4,-3)
" "C-=(-3,+8,-5)
" "vec(AC)=(-3-(-1))hati+(8-4)hatj+(-5-(-3))hatk
vec(AC)=-2hati-2hatj+4hatk

" "A-=(-1,+4,-3)
" "D-=(-3,+2,+1)
" "vec(AD)=(-3-(-1))hati+(2-4)hatj+(1-(-3))hatk
vec(AD)=-2hati-2hatj+4hatk

row1=+4,-2,-2
row2=-2,-2,+4
row3=-2,-2,+4
Since two rows in the determinant are same
determinant vanishes to zero.
Thus,
" "A-=(-1,+4,-3)
" "B-=(+3,+2,-5)
" "C-=(-3,+8,-5)
" "D-=(-3,+2,+1)
lie in a single plane