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 $\mathbf{u} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 1 \\ 4 \end{pmatrix}$.

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 $\mathbf{u} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 1 \\ 4 \end{pmatrix}$.

Show that $\mathbf{u} \times \mathbf{v}$ is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$ for $\mathbf{u} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 1 \\ 4 \end{pmatrix}$ 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 $\mathbf{u} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 1 \\ 4 \end{pmatrix}$.

Given $\mathbf{w} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$, calculate the scalar triple product $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$ for $\mathbf{u} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 1 \\ 4 \end{pmatrix}$. 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 $\mathbf{u} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 1 \\ 4 \end{pmatrix}$.

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

Given $\mathbf{a} = \mathbf{u} + 2\mathbf{v}$, calculate $\mathbf{u} \times \mathbf{a}$ for $\mathbf{u} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 1 \\ 4 \end{pmatrix}$.