IB Mathematics Analysis and Approaches 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 $\mathbf{a}=\left(\begin{array}{c}1\\ 4\\ -2\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}-2\\ 1\\ 3\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}3\\ 0\\ 1\end{array}\right)$, 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 $\frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{2}$ 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 $\mathbf{a}=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}2\\ 3\\ 4\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}5\\ 6\\ 7\end{array}\right)$, 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 $\mathbf{a}=\left(\begin{array}{c}1\\ 1\\ 3\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$, find the distance between $\mathbf{a}+\mathbf{b}+\frac{1}{2}\mathbf{c}$ and $-2(\mathbf{a}+\mathbf{b}+\mathbf{c})$.

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