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Question

If the vectors  $$- 3 \hat { \imath } + 4 \hat { \jmath } - 2 \hat { k } , \hat { \imath } + 2 \hat { k }$$  and  $$\hat { \imath } - p \hat { \jmath }$$  are coplanar, then find the value of  $$p .$$


Solution

We have,
$$\begin{array}{l} -3\hat { i } +4\hat { j } -2\hat { k } ,\, \, \, \, \, \hat { i } +2\hat { k } ,\, \, \hat { i } -9\hat { j }  \\ \left| { \begin{array} { *{ 20 }{ c } }{ -3 } & 4 & { -2 } \\ 1 & 0 & 2 \\ 1 & { -p } & 0 \end{array} } \right| =0 \\ -3\left( { 2p } \right) -4\left( { 0-2 } \right) -2\left( { -p } \right) =0 \\ -4p+8=0 \\ \therefore p=2 \end{array}$$

Hence, this is the answer.

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