Question Type 8: Determining components of a vector not acting or acting perpendicularly on another vector Exercises

Given $\mathbf{u}=(1,0,2)$, $\mathbf{v}=(0,1,3)$ and $\mathbf{w}=(2,1,1)$, calculate the scalar triple product $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$ and hence determine the volume of the parallelepiped defined by $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$.

Using generic components $\mathbf{u} = (u_1, u_2, u_3)$ and $\mathbf{v} = (v_1, v_2, v_3)$, prove that $\mathbf{u} \times \mathbf{v} = -\mathbf{v} \times \mathbf{u}$.

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

Given $\mathbf{u} = (2, -1, 3)$ and $\mathbf{v} = (0, 4, -2)$, compute $\mathbf{u} \times \mathbf{v}$ and verify that it is orthogonal to both $\mathbf{u}$ and $\mathbf{v}$.

Let $\mathbf{u}=(1,2,3)$, $\mathbf{v}=(0,1,4)$ and $\mathbf{w}=(2,0,1)$. Decompose $\mathbf{w}$ into a component perpendicular to the plane spanned by $\mathbf{u}$ and $\mathbf{v}$ and a component parallel to that plane.

Show that for any vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^3$, the squared magnitude of their cross product satisfies $|\mathbf{u}\times\mathbf{v}|^2 = |\mathbf{u}|^2\,|\mathbf{v}|^2 - (\mathbf{u}\cdot\mathbf{v})^2.$

If two vectors $\mathbf{u}$ and $\mathbf{v}$ have magnitudes $|\mathbf{u}|=5$, $|\mathbf{v}|=7$ and the angle between them is $60^\circ$, compute $|\mathbf{u}\times\mathbf{v}|$.

Show that $\mathbf{u} \times \mathbf{v} = \mathbf{0}$ and conclude the relationship between $\mathbf{u}$ and $\mathbf{v}$.

Given the vectors $\mathbf{u} = (1, 0, 1)$, $\mathbf{v} = (2, 1, 3)$ and $\mathbf{w} = (3, 2, 5)$, show that these vectors are coplanar.

Give an explicit example of three vectors $\mathbf{u},\mathbf{v},\mathbf{w}$ such that $(\mathbf{u}\times\mathbf{v})\times\mathbf{w}\neq\mathbf{u}\times(\mathbf{v}\times\mathbf{w})$, and compute both sides.

Given $\mathbf{u}=(3,1,0)$ and $\mathbf{v}=(1,2,2)$, find the area of the parallelogram spanned by $\mathbf{u}$ and $\mathbf{v}$.

Prove that for any nonzero vectors $\mathbf{u}, \mathbf{v}$ in $\mathbb{R}^3$, $(\mathbf{u} \times \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v}) = 0.$