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Show that the four points A, B, C and D with position vectors \(4\hat{i}+5\hat{j}+\hat{k}\) \(-\hat{j}-\hat{k},\) \(3\hat{i}+9\hat{j}+4\hat{k}\) and \(4(-\hat{i}+\hat{j}+\hat{k})\) respectively are coplanar.

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Position vector of \(A\equiv 4\hat{i}+5\hat{j}+\hat{k};\)

Position vector of \(B\equiv -\hat{j}-\hat{k}\)

Position vector of \(C\equiv 3\hat{i}+9\hat{j}+4\hat{k};\)

Position vector of \(D\equiv -4\hat{i}+4\hat{j}+4\hat{k}\)

Hence\(\vec{AB},\vec{AC}\) and \(\vec{AD}\) are coplanar i.e., points A, B, C, D are coplanar.

[Note: Three vectors \(\vec{a},\vec{b},\vec{c}\) are coplanar, if the scalar triple product of these three vectors is zero.]

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