Question Type 6: Finding the area of a parallelogram or triangle spanned by two vectors Exercises

Find the area of the triangle with vertices at the origin, the point with position vector $\mathbf{u}=(2,1,2)$ and the point with position vector $\mathbf{v}=(1,2,1)$.

Calculate the area of the parallelogram defined by $\mathbf{u}=(2,1,2)$ and $\mathbf{v}=(1,2,1)$.

Find the area of the triangle with vertices at the origin, $2\mathbf{u}$ and $3\mathbf{v}$, where $\mathbf{u}=(2,1,2)$ and $\mathbf{v}=(1,2,1)$.

Compute the area of the parallelogram defined by the vectors $3\mathbf{u}$ and $2\mathbf{v}$, where $\mathbf{u}=(2,1,2)$ and $\mathbf{v}=(1,2,1)$.

Determine the angle $\theta$ between $\mathbf{u}=(2,1,2)$ and $\mathbf{v}=(1,2,1)$.

Find the area of the parallelogram spanned by $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-\mathbf{v}$, where $\mathbf{u}=(2,1,2)$ and $\mathbf{v}=(1,2,1)$.

The points $A,B,C$ have position vectors $\mathbf{u}=(2,1,2)$, $\mathbf{v}=(1,2,1)$ and $\mathbf{u}+\mathbf{v}$. Find the area of triangle $ABC$.

Find the height of the parallelogram spanned by $\mathbf{u}=(2,1,2)$ and $\mathbf{v}=(1,2,1)$, measured perpendicular to the base vector $\mathbf{u}$.

Using the scalar product, verify the area of the parallelogram spanned by $\mathbf{u}=(2,1,2)$ and $\mathbf{v}=(1,2,1)$ without computing the cross product.

Find a unit normal vector to the plane of the parallelogram spanned by $\mathbf{u}=(2,1,2)$ and $\mathbf{v}=(1,2,1)$.

Determine the area of the quadrilateral with vertices $O$, $\mathbf{u}$, $\mathbf{u}+2\mathbf{v}$ and $2\mathbf{v}$ in that order, where $\mathbf{u}=(2,1,2)$ and $\mathbf{v}=(1,2,1)$.