IB Mathematics Analysis and Approaches 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 $$M=\left(\begin{array}{ccc}1& 0& 2\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).$$ 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 $\mathbf{a}=\left(\begin{array}{c}2\\ 1\\ 2\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}1\\ 2\\ 1\end{array}\right)$.

Find the area of the projection of the parallelogram spanned by $\mathbf{a}=\left(\begin{array}{c}2\\ 1\\ 2\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}1\\ 2\\ 1\end{array}\right)$ onto the $xy$-plane.