Question Type 5: Finding the cross product for two given vectors Exercises

Express the cross product $u\times v$ using the determinant of a $3\times3$ matrix for $u=(2,3,1)$ and $v=(4,1,4)$.

Compute the cross product of $u=(2,3,1)$ and $v=(4,1,4)$.

Compute $v\times u$ and verify the relationship between $v\times u$ and $u\times v$ for $u=(2,3,1)$ and $v=(4,1,4)$.

Find the area of the parallelogram spanned by $u=(2,3,1)$ and $v=(4,1,4)$.

Compute $(3u)\times(2v)$ where $u=(2,3,1)$ and $v=(4,1,4)$.

Show that $u\times v$ is perpendicular to both $u$ and $v$ for $u=(2,3,1)$ and $v=(4,1,4)$ by computing dot products.

Find the area of the triangle formed by $u=(2,3,1)$ and $v=(4,1,4)$ at the origin.

If $a=u+2v$, compute $u\times a$ for $u=(2,3,1)$ and $v=(4,1,4)$.

Verify the distributive property by computing $u\times(v+w)$ for $u=(2,3,1)$, $v=(4,1,4)$ and $w=(1,0,2)$.

Find the equation of the plane through the origin with normal vector $u\times v$, where $u=(2,3,1)$ and $v=(4,1,4)$.

Determine a unit vector perpendicular to both $u=(2,3,1)$ and $v=(4,1,4)$.

Given $w=(1,2,3)$, compute the scalar triple product $u\cdot(v\times w)$ for $u=(2,3,1)$ and $v=(4,1,4)$, and interpret its absolute value geometrically.