Show that the vectors \hat{i}-2\hat{j}+3\hat{k},-2\hat{i}+3\hat{j}-4\hat{k}\: and\: \hat{i}-3\hat{j}+5\hat{k}  are coplanar.

 

 

 

 
 
 
 
 

Answers (1)

Let    \vec{a}= \hat{i}-2\hat{j}+3\hat{k}
         \vec{b}= -2\hat{i}+3\hat{j}-4\hat{k}
         \vec{c}= \hat{i}-3\hat{j}+5\hat{k}
 \left [ \vec{a}\, \vec{b}\: \vec{c} \right ]=\begin{vmatrix} 1& -2 & 3\\ -2 & 3 &-4 \\ 1& -3 &5 \end{vmatrix}
             = 1\left ( 15-12 \right )+2\left ( -10+4 \right )+3\left ( 6-3 \right )
             = 3-12+9
             = 0
therefore a,b,c are coplanar.   

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