Question Type 4: Finding the area of a parallelogram or triangle given two vectors Exercises

Compute the area of the triangle with vertices at the origin, $\mathbf{a}=(2, 1, 2)$ and $\mathbf{b}=(1, 2, 1)$.

Compute the area of the triangle spanned by $3\mathbf{a}$ and $4\mathbf{b}$ where $\mathbf{a}=(2,1,2)$ and $\mathbf{b}=(1,2,1)$.

The linear transformation $T:\mathbb R^3\to\mathbb R^3$ has matrix

Find the area of the image of the parallelogram spanned by $\mathbf{a}=(2,1,2)$ and $\mathbf{b}=(1,2,1)$ under $T$.

Compute the area of the parallelogram spanned by $2\mathbf{a}$ and $3\mathbf{b}$, where $\mathbf{a}=(2,1,2)$ and $\mathbf{b}=(1,2,1)$.

Compute the area of the parallelogram spanned by $\mathbf{a}=(2,1,2)$ and $\mathbf{b}=(1,2,1)$.

Given points $P(0,0,0)$, $Q(2,1,2)$ and $R(1,2,1)$, compute the area of triangle $PQR$.

Let $\mathbf{c}=\mathbf{a}+\mathbf{b}$ and $\mathbf{d}=\mathbf{a}-\mathbf{b}$ for $\mathbf{a}=(2,1,2)$, $\mathbf{b}=(1,2,1)$. Compute the area of the parallelogram spanned by $\mathbf{c}$ and $\mathbf{d}$.

Find the acute angle between $\mathbf{a}=(2,1,2)$ and $\mathbf{b}=(1,2,1)$, then compute the area of the parallelogram these vectors span.

Compute the area of the parallelogram whose diagonals are given by $\mathbf{a}=\begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$.

Find the area of the projection of the parallelogram spanned by $\mathbf{a}=\begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$ onto the $xy$-plane.