Exercise · ZW-CROSS-PRODUCT-001 · ★☆☆ Basic
Compute \(\langle1,2,3\rangle\times\langle4,5,6\rangle\).
Choose one option.
Show answer
- B. \(\langle-3,6,-3\rangle\)
Explanation
\(\mathbf i\) component: \(2\cdot6-3\cdot5=-3\); \(\mathbf j\) component: \(-(1\cdot6-3\cdot4)=6\); \(\mathbf k\) component: \(1\cdot5-2\cdot4=-3\). The middle term of the expansion carries a minus sign.
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-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|?
- ZW-CROSS-PRODUCT-009 If mathbf u×mathbf v=langle1,2,2rangle, what is mathbf u×(2mathbf v+3mathbf u)?