Are the four points \(A(1,-1,1)\), \(B(-1,1,1)\), \(C(1,1,1)\) and \(D(2,-3,4)\) co-planar? Justify your answer.

Q.16 Are the four points \(A(1,-1,1)\), \(B(-1,1,1)\), \(C(1,1,1)\) and \(D(2,-3,4)\) co-planar? Justify your answer.

Doubtly Asked 29 Oct 2026 Board exam · July 2024 hsc 12th Save WhatsApp Report

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Answer: No, the points are not co-planar.

\(\overline{AB}=\bar b-\bar a=-2\hat i+2\hat j+0\hat k\), \(\overline{AC}=0\hat i+2\hat j+0\hat k\), \(\overline{AD}=\hat i-2\hat j+3\hat k\).

The points are co-planar iff \([\overline{AB}\ \overline{AC}\ \overline{AD}]=0\).

\([\overline{AB}\ \overline{AC}\ \overline{AD}]=\begin{vmatrix}-2&2&0\\0&2&0\\1&-2&3\end{vmatrix}=-2(6-0)-2(0-0)+0=-12\ne0\).

Since the scalar triple product is not zero, the vectors are not co-planar, so A, B, C, D are not co-planar.

Doubtly AI · 29 Oct 2026 AI-generated · not yet verified Report

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