Exercise · ZW-CROSS-PRODUCT-010 · ★★★ Advanced
If \(\mathbf u\times\mathbf v=\langle1,2,2\rangle\), what is the area of the parallelogram spanned by \(\mathbf v\) and \(\mathbf u\)?
Choose one option.
Show answer
- C. \(3\)
Explanation
Swapping the factors only reverses the vector, not its length: \(|\mathbf v\times\mathbf u|=|\langle1,2,2\rangle|=\sqrt{1+4+4}=3\). An area is never negative.
Review the concept: Cross product
The cross product of two vectors \(\mathbf u\) and \(\mathbf v\) in \(\mathbb R^3\) is the vector perpendicular to both whose length is the area of the parallelogram they span, oriented by the right-hand rule.
\[\mathbf u\times\mathbf v=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\u_1&u_2&u_3\\v_1&v_2&v_3\end{vmatrix}\]
More practice problems: cross product
- ZW-CROSS-PRODUCT-001 Compute langle1,2,3rangle×langle4,5,6rangle.
- ZW-CROSS-PRODUCT-002 Compute mathbf j×mathbf i.
- ZW-CROSS-PRODUCT-003 Compute langle2,0,1rangle×langle1,3,0rangle.
- ZW-CROSS-PRODUCT-004 What is the area of the parallelogram spanned by langle1,2,0rangle and langle3,1,0rangle?
- ZW-CROSS-PRODUCT-005 Find a vector perpendicular to both langle1,1,0rangle and langle0,1,1rangle.
- ZW-CROSS-PRODUCT-006 If mathbf u and mathbf v are nonzero and mathbf u×mathbf v=mathbf 0, then:
- ZW-CROSS-PRODUCT-007 Compute the area of the triangle with vertices (0,0,0), (2,1,0) and (1,3,0).
- ZW-CROSS-PRODUCT-008 If |mathbf u|=3, |mathbf v|=4 and the angle between them is 30^∘, what is |mathbf u×mathbf v|?