Cross Product — Question 7

Question 7 – Solution

Keep the component order and signs organized when expanding the determinant. After computing the cross product, use a dot-product check or the relevant magnitude formula to interpret it.

Use vectors from $A$: $\overrightarrow{AB}=\langle2,-1,4\rangle$, $\overrightarrow{AC}=\langle-2,3,1\rangle$, $\overrightarrow{AD}=\langle4,-4,7\rangle$.

Compute $\overrightarrow{AB}\times\overrightarrow{AC}=\langle-13,-10,4\rangle$.

Dot with $\overrightarrow{AD}$: $-52+40+28=16\ne0$. Therefore the points are $\boxed{\text{not coplanar}}$.

The result follows from the defining vector formulas used above, and each component, magnitude, or scalar condition has been checked against the information in the question.

The cross product produces a vector perpendicular to both inputs. Its direction follows the right-hand rule, while its magnitude records the area of the spanned parallelogram.