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Question 193 of 215

Q.Show that the points A(2, 1, -1), B(0, -1, 0), C(4, 0, 4) and D(2, 0, 1) are coplanar.

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2020Subjective· 3mImportance★★★★★
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Four points are coplanar iff the scalar triple product of any three vectors from a common point among them is zero.

AB⃗=B−A=(−2,−2,1)\vec{AB} = B-A = (-2,-2,1)

AC⃗=C−A=(2,−1,5)\vec{AC} = C-A = (2,-1,5)

AD⃗=D−A=(0,−1,2)\vec{AD} = D-A = (0,-1,2)

AC⃗×AD⃗=∣i^j^k^2−150−12∣=i^(−2+5)−j^(4−0)+k^(−2−0)=(3,−4,−2)\vec{AC}\times\vec{AD} = \begin{vmatrix}\hat i & \hat j & \hat k\\ 2 & -1 & 5\\ 0 & -1 & 2\end{vmatrix} = \hat i(-2+5) - \hat j(4-0) + \hat k(-2-0) = (3,-4,-2)

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