Cross Product — Question 4

Question 4

For what values of $k$ are $\vec u=\langle1,k,2\rangle$ and $\vec v=\langle2,2k,4\rangle$ parallel? What is their cross product then?

Question 4 – 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.

Observe directly that $\vec v=2\vec u$ for every real $k$.

Therefore the vectors are parallel for $\boxed{\text{all }k\in\mathbb R}$.

The cross product of parallel vectors is the zero vector, so $\boxed{\vec u\times\vec v=\vec0}$ for every $k$.

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.