Question Type 5: Visually representing vector operations Exercises

Given $\mathbf{a}=\begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$ and $\mathbf{c}=\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$, 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 $\mathbf{a}=\begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$ and $\mathbf{c}=\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$, compute the vector $\mathbf{b}-\mathbf{a}+\mathbf{c}$.

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