‹ Back to the paper
Prove the following.
Show that the four points A, B, C and D with position vectors 4îm̂ât̂ĥ + 5ĵm̂ât̂ĥ + k̂…
Show that the four points $A, B, C$ and $D$ with position vectors $4\hat{\imath} + 5\hat{\jmath} + \hat{k}$, $-\hat{\jmath} - \hat{k}$, $3\hat{\imath} + 9\hat{\jmath} + 4\hat{k}$ and $4(-\hat{\imath} + \hat{\jmath} + \hat{k})$ respectively, are coplanar.
Answer
Answer
The scalar triple product $[\overrightarrow{AB}\ \overrightarrow{AC}\ \overrightarrow{AD}] = 0$, so the four points are coplanar.
- $\overrightarrow{AB} = (0-4)\hat\imath + (-1-5)\hat\jmath + (-1-1)\hat k = -4\hat\imath - 6\hat\jmath - 2\hat k$
- $\overrightarrow{AC} = (3-4)\hat\imath + (9-5)\hat\jmath + (4-1)\hat k = -\hat\imath + 4\hat\jmath + 3\hat k$
- $\overrightarrow{AD} = (-4-4)\hat\imath + (4-5)\hat\jmath + (4-1)\hat k = -8\hat\imath - \hat\jmath + 3\hat k$
- Four points are coplanar if $[\overrightarrow{AB}\ \overrightarrow{AC}\ \overrightarrow{AD}] = 0$
- $\begin{vmatrix} -4 & -6 & -2 \\ -1 & 4 & 3 \\ -8 & -1 & 3 \end{vmatrix} = -4(12+3) + 6(-3+24) - 2(1+32)$, expanding along $R_1$
- $= -60 + 126 - 66 = 0$
- Hence $A, B, C, D$ are coplanar