LinAlg4.2.16

From Exampleproblems

Show that the four points having position vectors $6\bar{i}-7\bar{j},16\bar{i}-19\bar{j}-4\bar{k},3\bar{i}-6\bar{k},2\bar{i}+5\bar{j}+10\bar{k}\,$ are not coplanar.

Let $A=6\bar{i}-7\bar{j},B=16\bar{i}-19\bar{j}-4\bar{k},C=3\bar{i}-6\bar{k},D=2\bar{i}+5\bar{j}+10\bar{k}\,$ be the given vectors.

Then $\bar{AB}=10\bar{i}-12\bar{j}-4\bar{k},\bar{AD}=-4\bar{i}+12\bar{j}+10\bar{k},\bar{AC}=-6\bar{i}+10\bar{j}-6\bar{k}\,$

Now $[\bar{AB} \bar{AC} \bar{AD}]=\begin{vmatrix} 10 & -12 & -4 \\ -6 & 10 & -6 \\ -4 & 12 & 10 \end{vmatrix}=10(100+72)+12(-60-24)-4(-72+40)=1720-1008+128\,$ which is not equal to zero.