IB Mathematics Analysis and Approaches Question Type 2: Finding the Vector Equation of a Line Defined by Composite Vectors Exercises

Let $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}8\\ 1\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$.

Find the vector equation of the line passing through the points $P$ and $Q$, where $P=\mathbf{a}+\mathbf{b}$ and $Q=\mathbf{c}-2\mathbf{a}$.

Let $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}8\\ 1\\ 1\end{array}\right)$. Find the vector equation of the line passing through points $P$ and $Q$, where the position vector of $P$ is $2\mathbf{a}-\mathbf{b}$ and the position vector of $Q$ is $\mathbf{a}+2\mathbf{b}$.

Let $\mathbf{a}=(1,3,4)$, $\mathbf{b}=(8,1,1)$ and $\mathbf{c}=(0,1,0)$. Find the vector equation of the plane containing the points $P=\mathbf{a}+\mathbf{b}$, $Q=\mathbf{c}-2\mathbf{a}$ and $R=\mathbf{a}+2\mathbf{b}+2\mathbf{c}$.

Let $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}8\\ 1\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$.

Find the vector equation of the plane through the point $P$ with position vector $\mathbf{p}=2\mathbf{a}-\mathbf{b}+\mathbf{c}$ and parallel to both $\mathbf{u}=\mathbf{b}+\mathbf{a}$ and $\mathbf{v}=\mathbf{c}-\mathbf{a}$.

Let $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}8\\ 1\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$. Find the vector equation of the plane through $P=\mathbf{a}+\mathbf{b}-\mathbf{c}$, $Q=2\mathbf{a}+3\mathbf{b}$ and $R=4\mathbf{c}$.

Let $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}8\\ 1\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$.

Find the vector equation of the line passing through the point $P$ with position vector $\mathbf{p}=\mathbf{c}-\mathbf{a}$ and parallel to the vector $\mathbf{v}=\mathbf{b}+\mathbf{c}$.

Let $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}8\\ 1\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$.

Find the vector equation of the line passing through the point $P$ with position vector $\mathbf{p}=\mathbf{a}+2\mathbf{b}$, parallel to the vector $\mathbf{v}=\mathbf{c}-3\mathbf{b}+\mathbf{a}$.

Consider the points $A(1,3,4)$, $B(8,1,1)$ and $C(0,1,0)$.

Find the vector equation of the line passing through $A$ and $B$.

Let $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}8\\ 1\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$.

Find the vector equation of the line passing through the point $P$, with position vector $\overrightarrow{OP}=\mathbf{a}+2\mathbf{c}$, and parallel to the vector $\mathbf{v}=\mathbf{b}-3\mathbf{c}$.

Let $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}8\\ 1\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$. Find a vector equation of the plane passing through the points $P$, $Q$ and $R$, defined by $P=2\mathbf{a}-\mathbf{b}$, $Q=3\mathbf{b}+\mathbf{c}$ and $R=4\mathbf{c}-\mathbf{a}$.

Consider the points $A(1,3,4)$, $B(8,1,1)$ and $C(0,1,0)$. Find the vector equation of the plane passing through $A$, $B$ and $C$.

Let $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}8\\ 1\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$.

Find a vector equation of the plane passing through points $P$, $Q$ and $R$ defined by the position vectors: $$\overrightarrow{OP}=\mathbf{a}-2\mathbf{b}+\mathbf{c},\overrightarrow{OQ}=3\mathbf{a}+\mathbf{b}-\mathbf{c},\overrightarrow{OR}=2\mathbf{b}+2\mathbf{c}-\mathbf{a}$$