1
$\begingroup$

How can I prove that the points $A=(a_1, a_2, a_3)$, $B=(b_1, b_2, b_3)$, $C=(c_1, c_2, c_3)$, $D=(d_1, d_2, d_3)$ belong to the same plane? And if they do belong to the same plane, how can that plane be found?

(Original question included "I know how to prove three given points belong the same plane.".)

$\endgroup$
3
  • 2
    $\begingroup$ Three given points always belong to the same plane. $\endgroup$
    – TonyK
    Dec 2, 2015 at 21:20
  • $\begingroup$ Do you know determinants? $\endgroup$
    – Wojowu
    Dec 2, 2015 at 21:25
  • $\begingroup$ Yes. I now the determinants $\endgroup$
    – he hehi
    Dec 2, 2015 at 21:26

2 Answers 2

3
$\begingroup$

Hint: Translate by $-A$ so that we can assume $A=(0,0,0)$. Then points $A,B,C,D$ lie on a plane iff vectors $B,C,D$ do not span whole space, i.e. if $$\left| \begin{array}{ccc} b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ d_1 & d_2 & d_3 \end{array} \right|=0$$

$\endgroup$
1
  • $\begingroup$ Thanks. Can you tell me and one questions: how to find a dictance for the given two line l_1: 13x-y+6z+6=0; l_2: 26x-2y+12z-4=0 $\endgroup$
    – he hehi
    Dec 2, 2015 at 21:40
1
$\begingroup$

Take any three of the points and determine the equation of the plane. As TonyK said, three points always belong to one plane and, if they do not all lie in a line, then the determine a unique plane. Once you have the equation of the plane, put the coordinates of the fourth point into the equation to see if it is satisfied. If the three points you chose do happen to lie on a single line then you are done- any fourth point will determine a plane that all four points lie on.

$\endgroup$
2
  • $\begingroup$ For example : A(0,2,4); B((5,1,2); C(3,8,3); D(2,-2,1) how to proced $\endgroup$
    – he hehi
    Dec 2, 2015 at 21:30
  • 1
    $\begingroup$ @hehehi In your example, points A, B, and C determine a plane. Find the equation of that plane. Do the coordinates of D satisfy that equation? $\endgroup$
    – David K
    Dec 2, 2015 at 21:55

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .