Question Type 6: Finding the distance between more difficult position vectors Exercises

Find the distance between $\mathbf{a}-\mathbf{c}$ and $\mathbf{b}+2\mathbf{c}$.

Let $\mathbf{a}=(0,2,4)$, $\mathbf{b}=(5,1,-1)$ and $\mathbf{c}=(2,-2,3)$. Find the distance between $3\mathbf{a}-\frac{1}{2}\mathbf{b}+\mathbf{c}$ and $\mathbf{a}+\mathbf{b}-2\mathbf{c}$.

Let $\mathbf{a}=(3,3,3)$, $\mathbf{b}=(-1,0,2)$ and $\mathbf{c}=(0,1,-1)$. Find the distance between $\mathbf{a}-\mathbf{b}+2\mathbf{c}$ and $-3\mathbf{a}+\mathbf{b}-\mathbf{c}$

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

Given $\mathbf{a}=\begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} -2 \\ 1 \\ 3 \end{pmatrix}$ and $\mathbf{c}=\begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix}$, find the distance between the vectors $\mathbf{a}+\mathbf{b}-\mathbf{c}$ and $-\mathbf{a}+2\mathbf{b}+\mathbf{c}$.

Let $\mathbf{a}=(4,1,0)$, $\mathbf{b}=(0,-3,2)$ and $\mathbf{c}=(5,1,-1)$. Find the distance between $\frac{2\mathbf{a}+\mathbf{b}-\mathbf{c}}{3}$ and $\mathbf{a}-2\mathbf{b}+\frac{1}{2}\mathbf{c}$.

Given $\mathbf{a}=(1,2,3)$, $\mathbf{b}=(4,-1,2)$ and $\mathbf{c}=(0,1,1)$, find the distance between $\tfrac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{2}$ and $\mathbf{b}-2\mathbf{c}+\mathbf{a}$.

Let $\mathbf{a}=(2,-1,4)$, $\mathbf{b}=(0,3,1)$ and $\mathbf{c}=(-1,2,2)$. Find the distance between $2\mathbf{a}-\mathbf{b}+\mathbf{c}$ and $\mathbf{a}+3\mathbf{b}-\frac{1}{4}\mathbf{c}$.

Given $\mathbf{a}=\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}$ and $\mathbf{c}=\begin{pmatrix} 5 \\ 6 \\ 7 \end{pmatrix}$, find the distance between the vectors $\mathbf{a}+\mathbf{b}+\mathbf{c}$ and $4\mathbf{a}-\mathbf{b}-2\mathbf{c}$.

Given the standard basis $\mathbf{a}=(1,0,0)$, $\mathbf{b}=(0,1,0)$, $\mathbf{c}=(0,0,1)$, find the distance between $\mathbf{a}+\mathbf{b}+\mathbf{c}$ and $-(\mathbf{a}-\mathbf{b}+2\mathbf{c})$.

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 distance between $\mathbf{a}+\mathbf{b}+\frac{1}{2}\mathbf{c}$ and $-2(\mathbf{a}+\mathbf{b}+\mathbf{c})$.

Let $\mathbf{a}=\begin{pmatrix} -2 \\ 0 \\ 1 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 3 \\ 2 \\ -1 \end{pmatrix}$ and $\mathbf{c}=\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$. Find the distance between the point defined by $-\mathbf{a}+\frac{3}{2}\mathbf{b}-\mathbf{c}$ and the point defined by $2\mathbf{a}-\mathbf{b}+\frac{1}{2}\mathbf{c}$.