Explain how you would prove if four given points are coplanar. Use your method to determine if A (3, 4, -2), B (8, 5, 0), C (1, 10, -6) and D (9, 2, 2) are coplanar.

Question

unklerhaukus · Answer

with a matrix?

unklerhaukus · Answer

for 4 points to be co-planar , some of them must be linear combinations of others

anonymous · Answer

and how should i proceed please.

ganeshie8 · Answer

or you could work it by finding the distance between one of the points to the plane determined by remaining  3 points

ganeshie8 · Answer

are u familiar with cross product ?

anonymous · Answer

im not sure, im doing the xproduct right

anonymous · Answer

AB=(5,1,2)

AC=(-2,6,-4)
AD=(6,-2,-0)

for ABxAC ive got (5,-1,10)

Im stuck here dont know how to go foward

ganeshie8 · Answer

check your AD again

anonymous · Answer

yeah its (-6,-2,4)

ganeshie8 · Answer

A (3, 4, -2), B (8, 5, 0), C (1, 10, -6) and D (9, 2, 2) 

AB=(5,1,2)
AC=(-2,6,-4)

AD=(6,-2, 4)

work ABxAC again

ganeshie8 · Answer

$$\large AB\times AC = \left|\begin{array}{|c|c|c|} i&j&k\5&1&2\-2&6&-4\end{array}\right| = -16i + 16j +32k$$

ganeshie8 · Answer

so that the normal vector the plane determined by lines AB and AC

ganeshie8 · Answer

ABxAC = (-16, 16, 32)

dot this with AD, if both are perpendicular then the dot product will be 0 confirming that D lies in the same plane as A,B,C

ganeshie8 · Answer

AD . (AB x AC) =  (6,-2, 4) . (-16, 16, 32)
                       = ?

anonymous · Answer

yeah its zero

anonymous · Answer

thx for you help, I will go and review the cross product.. forgot everything about that

ganeshie8 · Answer

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