IB Mathematics Applications & Interpretation Question Type 5: Visually Representing Vector Operations Exercises

Given $\mathbf{a}=\left(\begin{array}{c}1\\ 1\\ 3\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$, find the vector $\mathbf{a}+\mathbf{b}-\mathbf{c}$.

Given $\mathbf{a}=(1,1,3)$, $\mathbf{b}=(1,0,-1)$ and $\mathbf{c}=(1,1,1)$, calculate the scalar triple product $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$ and interpret its absolute value geometrically.

Given $\mathbf{a}=\left(\begin{array}{c}1\\ 1\\ 3\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$, compute the vector $\mathbf{b}-\mathbf{a}+\mathbf{c}$.

Given $\mathbf{a}=\left(\begin{array}{c}1\\ 1\\ 3\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$, find the magnitude of the vector $\mathbf{a}+\mathbf{b}+\frac{1}{2}\mathbf{c}$.

Given $\mathbf{a}=(1,1,3)$, $\mathbf{b}=(1,0,-1)$ and $\mathbf{c}=(1,1,1)$, find the vector that goes from the point represented by $\mathbf{b}$ to the point represented by $\mathbf{a}+\mathbf{c}$.

Given $\mathbf{a}=(1,1,3)$, $\mathbf{b}=(1,0,-1)$ and $\mathbf{c}=(1,1,1)$, find the magnitude of the vector $-2(\mathbf{a}+\mathbf{b}+\mathbf{c})$.

Given $\mathbf{a}=(1,1,3)$, $\mathbf{b}=(1,0,-1)$ and $\mathbf{c}=(1,1,1)$, find $2\mathbf{a}-\mathbf{b}+3\mathbf{c}$.

Given $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$, compute the projection of $\mathbf{b}$ onto $\mathbf{c}$.

Given $\mathbf{a}=(1,1,3)$, $\mathbf{b}=(1,0,-1)$ and $\mathbf{c}=(1,1,1)$, find the midpoint of the segment joining the points represented by $\mathbf{b}$ and $2\mathbf{c}$.

Given $\mathbf{a}=(1,1,3)$, $\mathbf{b}=(1,0,-1)$ and $\mathbf{c}=(1,1,1)$, calculate the distance between the points represented by $\mathbf{u}=\mathbf{a}+\mathbf{b}+\frac{1}{2}\mathbf{c}$ and $\mathbf{v}=-2(\mathbf{a}+\mathbf{b}+\mathbf{c})$.

Given $\mathbf{a}=(1,1,3)$, $\mathbf{b}=(1,0,-1)$ and $\mathbf{c}=(1,1,1)$, find a unit vector in the direction of $\mathbf{a}+\mathbf{b}-\mathbf{c}$.

Given $\mathbf{a}=\left(\begin{array}{c}1\\ 1\\ 3\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$, compute $(\mathbf{a}+\mathbf{b})-(\mathbf{b}+\mathbf{c})$.