Question Type 2: Finding the vector equation of a line defined by composite vectors Exercises

Let $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 8 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

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 $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 8 \\ 1 \\ 1 \end{pmatrix}$. 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 $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 8 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

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 $\mathbf{a}=\begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 8 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{c}=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$. 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 $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 8 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

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 $\mathbf{a}=\begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 8 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{c}=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

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 $\mathbf{a}=\begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 8 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{c}=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

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

Let $\mathbf{a}=\begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 8 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{c}=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$. 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 $\mathbf{a}=\begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 8 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{c}=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

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