Find the value of x,for which the four points A\left ( x,-1,-1 \right ),\: B\left ( 4,5,1 \right ),\, C\left ( 3,9,4 \right )\: and\: D\left ( -4,4,4 \right )  are coplanar.

 

 

 

 
 
 
 
 

Answers (1)

We have \overline{BA}= \left ( x-4 \right )\hat{i}-6\hat{j}-2\hat{k}
               \overline{BC}= -\hat{i}+4\hat{j}+3\hat{k}
               \overline{BD}= -8\hat{i}-\hat{j}+3\hat{k}
If points A,B C and D are coplanar then \left | \overline{BA}\: \, \overline{BC}\: \, \overline{BD}\right |= 0
That  is  \begin{vmatrix} x-4 & -6 &-2 \\ -1& 4 & 3\\ -8& -1 & 3 \end{vmatrix}= 0
\Rightarrow 15\left ( x-4 \right )+6\times 21-2\times 33= 0
\Rightarrow 15x-60+126-66= 0
\therefore x= 0
 

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