IB Mathematics Applications & Interpretation Question Type 5: Finding the Cross Product for Two Given Vectors Exercises

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

Find the area of the triangle with vertices at the origin and the points with position vectors $\mathbf{u}=\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}4\\ 1\\ 4\end{array}\right)$.

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

Find the cross product $\mathbf{u}\times \mathbf{v}$ by expressing it as the determinant of a $3\times 3$ matrix, where $\mathbf{u}=\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}4\\ 1\\ 4\end{array}\right)$.

Show that $\mathbf{u}\times \mathbf{v}$ is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$ for $\mathbf{u}=\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}4\\ 1\\ 4\end{array}\right)$ by computing dot products.

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

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

Compute $(3\mathbf{u})\times (2\mathbf{v})$ where $\mathbf{u}=\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}4\\ 1\\ 4\end{array}\right)$.

Given $\mathbf{w}=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)$, calculate the scalar triple product $\mathbf{u}\cdot (\mathbf{v}\times \mathbf{w})$ for $\mathbf{u}=\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}4\\ 1\\ 4\end{array}\right)$. Interpret the absolute value of this result geometrically.

Find the equation of the plane through the origin with normal vector $\mathbf{u}\times \mathbf{v}$, where $\mathbf{u}=\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}4\\ 1\\ 4\end{array}\right)$.

Verify the distributive property $\mathbf{u}\times (\mathbf{v}+\mathbf{w})=\mathbf{u}\times \mathbf{v}+\mathbf{u}\times \mathbf{w}$ for the vectors $\mathbf{u}=\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$, $\mathbf{v}=\left(\begin{array}{c}4\\ 1\\ 4\end{array}\right)$, and $\mathbf{w}=\left(\begin{array}{c}1\\ 0\\ 2\end{array}\right)$.

Given $\mathbf{a}=\mathbf{u}+2\mathbf{v}$, calculate $\mathbf{u}\times \mathbf{a}$ for $\mathbf{u}=\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}4\\ 1\\ 4\end{array}\right)$.