Using vectors,find the value of x such that the four points A(x,5,-1),B(3,2,1),C(4,5,5) and D(4,2,-2) are coplanar.

 

 

 

 
 
 
 
 

Answers (1)

Here \overline{BA}= \left ( x-3 \right )\hat{i}+3\hat{j}-2\hat{k}
         \overline{BC}= \hat{i}+3\hat{j}+4\hat{k}
         \overline{BD}= \hat{i}-3\hat{k}
As A,B, C and D are coplanar so \left [ \overrightarrow{BA}\, \overrightarrow{BC} \, \overrightarrow{BD} \right ]
                           = \begin{vmatrix} x-3 &3 &-2 \\ 1& 3 & 4\\ 1& 0 & -3 \end{vmatrix}= 0    
\Rightarrow \left ( x-3 \right )\left ( -9 \right )-3\left ( -7 \right )-2\left ( -3 \right )= 0
                     \therefore x= 6

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