Vectors: Mathematics 9709 (Cambridge International AS & A Level)

The vector $\mathbf{p} = 4\mathbf{i} - \mathbf{j} + 8\mathbf{k}$.

Which of the following is the unit vector in the direction of $\mathbf{p}$?

Relative to an origin $O$, $\overrightarrow{OA} = \mathbf{a} = \begin{pmatrix}3\\-1\\2\end{pmatrix}, \qquad \overrightarrow{OB} = \mathbf{b} = \begin{pmatrix}1\\4\\-1\end{pmatrix}.$

(a) Find $\overrightarrow{AB}$. [2]

(b) Find $|\overrightarrow{AB}|$, giving your answer in the form $\sqrt{n}$ for an integer $n$. [2]

(c) Find angle $AOB$, the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$, correct to $1$ decimal place. [3]

Lines $l_1$ and $l_2$ have vector equations $l_1: \mathbf{r} = \begin{pmatrix}2\\1\\-1\end{pmatrix} + s\begin{pmatrix}4\\-2\\6\end{pmatrix}, \qquad l_2: \mathbf{r} = \begin{pmatrix}1\\-3\\2\end{pmatrix} + t\begin{pmatrix}-6\\3\\-9\end{pmatrix}.$

What is the relationship between $l_1$ and $l_2$?

Lines $l_1$ and $l_2$ have vector equations $l_1: \mathbf{r} = \begin{pmatrix}1\\2\\-3\end{pmatrix} + s\begin{pmatrix}2\\-1\\1\end{pmatrix}, \qquad l_2: \mathbf{r} = \begin{pmatrix}1\\-1\\2\end{pmatrix} + t\begin{pmatrix}1\\1\\-2\end{pmatrix}.$

(a) Show that $l_1$ and $l_2$ are not parallel. [2]

(b) Show that $l_1$ and $l_2$ intersect, and find the position vector of their point of intersection. [4]

(c) Find the acute angle between $l_1$ and $l_2$, correct to $1$ decimal place. [3]

Points $P$ and $Q$ have position vectors $\overrightarrow{OP} = \begin{pmatrix}2\\-1\\5\end{pmatrix}$ and $\overrightarrow{OQ} = \begin{pmatrix}6\\3\\-3\end{pmatrix}$.

(a) Find a vector equation for the line $PQ$, giving the direction vector in its simplest integer form. [3]

(b) The point $R(-2,-5,13)$ is claimed to lie on line $PQ$. Determine, showing full working, whether this is true. [3]

(c) Find the unit vector in the direction of $\overrightarrow{PQ}$. [2]

The vector $\mathbf{v} = 2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}$.

What is $|\mathbf{v}|$?

Points $M$ and $N$ have position vectors $\overrightarrow{OM} = \begin{pmatrix}2\\5\\-3\end{pmatrix}, \qquad \overrightarrow{ON} = \begin{pmatrix}5\\9\\9\end{pmatrix}$ relative to an origin $O$.

(a) Find $\overrightarrow{MN}$. [2]

(b) Find $|\overrightarrow{MN}|$. [2]

(c) Find the unit vector in the direction of $\overrightarrow{MN}$, and hence write down the unit vector in the direction of $\overrightarrow{NM}$. [3]

Vectors $\mathbf{a} = \begin{pmatrix}1\\2\\2\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}2\\-2\\1\end{pmatrix}$.

What is the angle between $\mathbf{a}$ and $\mathbf{b}$?

Points $A$, $B$ and $C$ have position vectors $\overrightarrow{OA} = \begin{pmatrix}1\\-2\\3\end{pmatrix}, \qquad \overrightarrow{OB} = \begin{pmatrix}3\\2\\1\end{pmatrix}, \qquad \overrightarrow{OC} = \begin{pmatrix}4\\-2\\7\end{pmatrix}$ relative to an origin $O$.

(a) Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$. [2]

(b) Use the scalar product to find angle $BAC$, correct to $1$ decimal place. [3]

(c) State, with a reason, whether angle $BAC$ is exactly $90^\circ$. [2]

Vectors $\mathbf{a} = \begin{pmatrix}3\\k\\-2\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}2\\-1\\4\end{pmatrix}$, where $k$ is a constant.

(a) Given that $\mathbf{a}$ and $\mathbf{b}$ are perpendicular, find the value of $k$. [3]

(b) Using this value of $k$, find $|\mathbf{a}|$, giving your answer in the form $\sqrt{n}$ for an integer $n$. [2]

(c) Find the unit vector in the direction of $\mathbf{a}$. [2]