Vectors

Express $\displaystyle \boldsymbol v$ in terms of $\displaystyle \boldsymbol u$ and $\displaystyle \boldsymbol w .$

$\displaystyle \overrightarrow{\textsf{PQ}}$ and $\displaystyle \overrightarrow{\textsf{RQ}}$ represent vectors $\displaystyle \boldsymbol u$ and $\displaystyle \boldsymbol v$ respectively.

(a)  Express $\displaystyle \overrightarrow{\textsf{RP}}$ in terms of $\displaystyle \boldsymbol u$ and $\displaystyle \boldsymbol v .$
(b)  Express $\displaystyle \overrightarrow{\textsf{RS}}$ in terms of $\displaystyle \boldsymbol u$ and $\displaystyle \boldsymbol v .$ Give your answer in its simplest form.

B is the point (8,6,0).

The apex D is directly above the centre of the base.

The height of the pyramid is 7 units.

(a) Express $\displaystyle \overrightarrow{\textsf{AC}}$ in component form.
(b) Express $\displaystyle \overrightarrow{\textsf{AD}}$ in component form.
(c) Determine the exact value of $\displaystyle \vert \overrightarrow{\textsf{AD}} \vert.$

The diagram below shows parallelogram ABCD.

$\displaystyle \vec{AB}$ represents vector $\displaystyle \mathbf{u}$ and $\displaystyle \vec{BC}$ represents vector $\displaystyle \mathbf{v}$.
Express $\displaystyle \vec{BD}$ in terms of $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$.

In the diagram below, $\displaystyle \overrightarrow{AB}$ and $\displaystyle \overrightarrow{EA}$ represent the vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{w}$ respectively.

$\displaystyle \bullet \quad \overrightarrow{ED}=2\overrightarrow{AB}$
$\displaystyle \bullet \quad \overrightarrow{EA}=2\overrightarrow{DC}$
Express $\displaystyle \overrightarrow{BC}$ in terms of $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{w}$.
Give your answer in its simplest form.

In triangle PQR, $\displaystyle \overrightarrow{PR}=\begin{pmatrix}6\\ -4\end{pmatrix}$ and $\displaystyle \overrightarrow{RQ}=\begin{pmatrix}-1\\ 8\end{pmatrix}.$

(a)  Express $\displaystyle \overrightarrow{PQ}$ in component form.

M is the midpoint of PR.
(b)  Express $\displaystyle \overrightarrow{MQ}$ in component form.

The vectors $\displaystyle \boldsymbol u$ and $\displaystyle \boldsymbol v$ are shown in the diagram below.

Find the resultant vector $\displaystyle \boldsymbol u - \boldsymbol v.$
Express your answer in component form.

The triangle ABC is shown below

$\displaystyle \overrightarrow{\textsf{AB}} =\boldsymbol u$ and $\displaystyle \overrightarrow{\textsf{AC}}=\boldsymbol t.$
G is the point such that CG = $\displaystyle \large\frac{1}{3}$CB.
Express $\displaystyle \overrightarrow{\textsf{AG}}$ in terms of $\displaystyle \boldsymbol u$ and $\displaystyle \boldsymbol t.$
Give your answer in simplest form.

Given $\displaystyle \mathbf{a}=\begin{pmatrix}3\\ 4\\ -1\end{pmatrix}$ and $\displaystyle \mathbf{b}=\begin{pmatrix}5\\ 3\\ 2\end{pmatrix}$, find the resultant vector $\displaystyle 3\mathbf{a}+\mathbf{b}.$
Express your answer in component form.