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  2. Let Then If c 2 = –1 and c 3 = 1, show that no value of c 1 can make coplanar.

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NCERT

Let Then If c 2 = –1 and c 3 = 1, show that no value of c 1 can make coplanar.

4(b)   Let \vec a = \hat i + \hat j + \hat k , \vec b = \hat i \: \: and \: \: \vec c = c_1 \hat i + c_2 \hat j + c_3 \hat k  Then
             If c_2 =- 1 and c_3 =1 show that no value of c_3 can make \vec a , \vec b , \vec c coplanar.    

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P Pankaj Sanodiya

Given

\vec a = \hat i + \hat j + \hat k , \vec b = \hat i \: \: and \: \: \vec c = c_1 \hat i - \hat j + \hat k

these will be coplanar when,

\left [ \vec a ,\vec b, \vec c \right ]=\begin{vmatrix} 1 &1 &1 \\ 1& 0& 0\\ c_1&-1 &1 \end{vmatrix}=1(0)-1(1)+1(-1)=-2

Hence the value of  \left [ \vec a ,\vec b, \vec c \right ]  is -2, irrespective of the value of c_3. Hence  no value of  c_3 satisfies the condition of three vectors being coplanar.

 

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