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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES5 Marks
Question
Show that four points A, B, C and D whose position vectors are $4\hat{\text{i}} + 5\hat{\text{j}} + \hat{\text{k}}, -\hat{\text{j}} - \hat{\text{k}}, 3\hat{\text{i}} + 9\hat{\text{j}} + \hat{\text{k}}$ and $4(-\hat{\text{i}} + \hat{\text{j}} + \hat{\text{k}})$ respectively are coplanar.
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Answer

Here
$\begin{matrix} \overrightarrow{\text{AB}} & = & -4\hat{\text{i}} - 6\hat{\text{j}} - 2\hat{\text{k}} \\ \overrightarrow{\text{AC}} & = & -\hat{\text{i}} + 4\hat{\text{j}} + 3\hat{\text{k}} \\ \overrightarrow{\text{AD}} & = & -8\hat{\text{i}} - \hat{\text{j}} + 3\hat{\text{k}} \end{matrix}$
For them to be coplanar, $\bigg[\overrightarrow{\text{AB}} \overrightarrow{\text{ AC }}\overrightarrow{\text{AD}}\bigg] = 0$
$\text{i.e}\begin{vmatrix} -4 & -6 & -2 \\ -1 & 4 & 3 \\ -8 & -1 & 3 \end{vmatrix} = -60 + 126 - 66 = 0 $
$\therefore$ Points A, B, C and D are coplanar.

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