Referred to a fixed origin $\displaystyle O$, the position vector of the point $\displaystyle A$ is $\displaystyle (9\mathbf{a}+12\mathbf{b})$

and the position vector of the point $\displaystyle B$ is $\displaystyle (2\mathbf{a} + p\mathbf{b})$ where $\displaystyle |\mathbf{a}| = |\mathbf{b}| = 1$

Given that $\displaystyle |AB| = 25$

(a) show that the two possible values of $\displaystyle p$ are $\displaystyle -12$ and $\displaystyle 36$

For $\displaystyle p = -12$

(b) find a unit vector parallel to $\displaystyle \overrightarrow{OB}$

Give your answer in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$ in its simplest form.

Point $\displaystyle C$ lies on $\displaystyle OB$ such that $\displaystyle OC : CB = 2 : 3$

For $\displaystyle p = 36$

(c) find $\displaystyle \overrightarrow{CA}$

Give your answer in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$ in its simplest form.

Figure 2 shows triangle $\displaystyle OAB$ with

(a) Find $\displaystyle \overrightarrow{AB}$ as a simplified expression in terms of $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$

Point $\displaystyle C$ is such that $\displaystyle AC$ is parallel to $\displaystyle OB$

Given that $\displaystyle \overrightarrow{OC} = \mu\mathbf{v}$

(b) find the value of $\displaystyle \mu$

The lines $\displaystyle OC$ and $\displaystyle AB$ intersect at point $\displaystyle X$

Point $\displaystyle D$ lies on $\displaystyle \overrightarrow{OC}$ such that

area of triangle $\displaystyle BOX$ : area of triangle $\displaystyle BXD = 2 : 3$

Using a vector method,

(c) find $\displaystyle \overrightarrow{BD}$ as a simplified expression in terms of $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$

The points $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ and $\displaystyle D$ are the vertices of a quadrilateral where

(a) Show that $\displaystyle ABCD$ is a trapezium.

Given that $\displaystyle |\mathbf{a}| = 1$ and $\displaystyle |\mathbf{b}| = 1$ and that $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$ are perpendicular to each other

(b) find a unit vector parallel to $\displaystyle \overrightarrow{BD}$

The point $\displaystyle Y$ lies on $\displaystyle CD$ such that $\displaystyle CY : YD = 1 : 2$

The lines $\displaystyle AC$ and $\displaystyle BY$ intersect at the point $\displaystyle X$

(c) (i) Use a vector method to find the ratio $\displaystyle BX : XY$

(ii) Hence find, in its simplest form, the ratio

area of triangle $\displaystyle CXY$ : area of trapezium $\displaystyle ABCD$

The points $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ and $\displaystyle D$ are the vertices of a quadrilateral $\displaystyle ABCD$ such that

Show that $\displaystyle ABCD$ is a parallelogram.

Figure 3 shows a trapezium $\displaystyle ABCD$

(a) Find $\displaystyle \overrightarrow{AB}$ as a simplified expression in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$

The diagonals $\displaystyle BD$ and $\displaystyle AC$ intersect at point $\displaystyle X$ where $\displaystyle \overrightarrow{BX}=k\overrightarrow{BD}$

(b) Using a vector method, find the value of $\displaystyle k$

(c) Find the ratio of the area of triangle $\displaystyle CXD$ : area of the trapezium $\displaystyle ABCD$

The points $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ and $\displaystyle D$ are the vertices of a quadrilateral such that

(a) Show that $\displaystyle ABCD$ is a parallelogram.

$\displaystyle BC$ is extended to the point $\displaystyle E$ such that $\displaystyle BCE$ is a straight line.

Point $\displaystyle F$ lies on $\displaystyle CD$ such that $\displaystyle CF : FD = 1 : 2$

Given that $\displaystyle A$, $\displaystyle F$ and $\displaystyle E$ are collinear,

(b) find the vector $\displaystyle \overrightarrow{AE}$ in the form $\displaystyle X\mathbf{a} + Y\mathbf{b}$ where $\displaystyle X$ and $\displaystyle Y$ are rational numbers to be found.

Figure 2 shows triangle $\displaystyle AOB$

(a) (i) Find $\displaystyle \overrightarrow{AB}$ in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$

(ii) Find, in its simplest form, the exact value of $\displaystyle |\overrightarrow{AB}|$

(b) Find the area of triangle $\displaystyle AOB$

The point $\displaystyle C$ lies on $\displaystyle AB$ and $\displaystyle OD$ such that $\displaystyle O$, $\displaystyle C$ and $\displaystyle D$ are collinear.

(c) Use a vector method to find vector $\displaystyle \overrightarrow{OC}$ as a simplified expression in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$

$\displaystyle O, A$ and $\displaystyle B$ are fixed points such that

Given that $\displaystyle a > 0$

(a) find the value of $\displaystyle a$

(b) Hence find a unit vector parallel to $\displaystyle \overrightarrow{OA}$

$\displaystyle O, A$ and $\displaystyle B$ are fixed points such that

The unit vector parallel to $\displaystyle \overrightarrow{OB}$ is $\displaystyle \frac{\sqrt{17}}{34} [(3a + 2)\mathbf{i} + b\mathbf{j}]$

Given that $\displaystyle a$ and $\displaystyle b$ are constants where $\displaystyle a > 0$ and $\displaystyle b > 0$

find the exact value of

(i) $\displaystyle a$

(ii) $\displaystyle b$

In Figure 5, $\displaystyle \overrightarrow{OA} = 2\mathbf{a}$, $\displaystyle \overrightarrow{OB} = 4\mathbf{b}$ and $\displaystyle M$ is the midpoint of $\displaystyle OA$.

The point $\displaystyle Y$ lies on $\displaystyle AB$ such that $\displaystyle AY : YB = 3 : 1$

The point $\displaystyle X$ lies on $\displaystyle OB$ produced.

(a) Find as simplified expressions in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$

(i) $\displaystyle \overrightarrow{AB}$      (ii) $\displaystyle \overrightarrow{MY}$

The points $\displaystyle M$, $\displaystyle Y$ and $\displaystyle X$ are collinear.

(b) Find the ratio $\displaystyle OB : OX$

(c) Find the ratio of $\displaystyle (\mathrm{Area} \Delta YBX) : (\mathrm{Area} \Delta OAX)$

Figure 3 shows quadrilateral $\displaystyle OABC$ where

The point $\displaystyle M$ is the midpoint of $\displaystyle OC$

(a) Find $\displaystyle \overrightarrow{MA}$ as a simplified expression in terms of $\displaystyle \mathbf{p}$ and $\displaystyle \mathbf{q}$

The point $\displaystyle N$ lies on $\displaystyle OB$ such that $\displaystyle M, N$ and $\displaystyle A$ are collinear.

(b) Find the ratio $\displaystyle MN : NA$

$\displaystyle O, A$ and $\displaystyle B$ are fixed points such that

(a) Find the possible values of $\displaystyle a$

Given that $\displaystyle a > 0$

(b) find a unit vector that is parallel to $\displaystyle \overrightarrow{AB}$

Figure 4 shows triangle $\displaystyle OAB$ in which

The point $\displaystyle P$ lies on $\displaystyle AB$ such that $\displaystyle AP:PB=3:1$.

The point $\displaystyle M$ is the midpoint of $\displaystyle OA$ and the point $\displaystyle N$ is the midpoint of $\displaystyle OP$.

(a) Find, as simplified expressions in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$, the vector

(i) $\displaystyle \overrightarrow{OP}$,

(ii) $\displaystyle \overrightarrow{MN}$.

The point $\displaystyle C$ lies on $\displaystyle OB$ such that $\displaystyle ANC$ is a straight line.

(b) Using a vector method, find $\displaystyle \overrightarrow{OC}$ as a simplified expression in terms of $\displaystyle \mathbf{b}$.

Given that

(c) find the exact value of $\displaystyle K$.

Figure 3 shows quadrilateral $\displaystyle ABCD$ such that

(a) Prove that $\displaystyle \overrightarrow{AB}$ is parallel to $\displaystyle \overrightarrow{DC}$.

The diagonals, $\displaystyle AC$ and $\displaystyle BD$, of the quadrilateral intersect at the point $\displaystyle Y$.

(b) Using a vector method, find $\displaystyle \overrightarrow{AY}$ as a simplified expression in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$.

$\displaystyle O$, $\displaystyle A$ and $\displaystyle B$ are fixed points such that

Given that

(a) find the value of $\displaystyle p$.

Using this value of $\displaystyle p$,

(b) find a unit vector that is parallel to $\displaystyle \overrightarrow{AB}$.

The position vector of the point $\displaystyle A$ is $\displaystyle 3\mathbf{i}-2\mathbf{j}$, referred to a fixed origin $\displaystyle O$.

The point $\displaystyle B$ is such that

(a) Find the position vector of $\displaystyle B$ as a simplified expression in terms of $\displaystyle \mathbf{i}$ and $\displaystyle \mathbf{j}$.

(b) Find the magnitude of $\displaystyle \overrightarrow{AB}$.

(c) Find a unit vector, in terms of $\displaystyle \mathbf{i}$ and $\displaystyle \mathbf{j}$, that is parallel to $\displaystyle \overrightarrow{AB}$.

Figure 1 shows triangle $\displaystyle OAB$ and triangle $\displaystyle OCD$.

(a) Find $\displaystyle \overrightarrow{DC}$ as a simplified expression in terms of $\displaystyle \mathbf{p}$ and $\displaystyle \mathbf{q}$.

The line $\displaystyle DC$ meets the line $\displaystyle AB$ at $\displaystyle F$.

(b) Using a vector method, find $\displaystyle \overrightarrow{OF}$ as a simplified expression in terms of $\displaystyle \mathbf{p}$ and $\displaystyle \mathbf{q}$.

The point $\displaystyle G$ lies on $\displaystyle OB$ such that $\displaystyle FG$ is parallel to $\displaystyle AO$.

(c) Using a vector method, find $\displaystyle \overrightarrow{OG}$ as a simplified expression in terms of $\displaystyle \mathbf{p}$ and $\displaystyle \mathbf{q}$.

In Figure 5,

The point $\displaystyle C$ divides $\displaystyle OB$ in the ratio $\displaystyle 1:3$. The point $\displaystyle D$ is the midpoint of $\displaystyle AC$.

(a) Find, as a simplified expression in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$,

(i) $\displaystyle \overrightarrow{AC}$,

(ii) $\displaystyle \overrightarrow{OD}$,

(iii) $\displaystyle \overrightarrow{BD}$.

The point $\displaystyle E$ is such that $\displaystyle \overrightarrow{OE}=\lambda\overrightarrow{OA}$. Given that $\displaystyle E$, $\displaystyle D$ and $\displaystyle B$ are collinear,

(b) find the value of $\displaystyle \lambda$.

Given that

(c) find the value of $\displaystyle \mu$.

Figure 1 shows a triangle $\displaystyle OXY$.

$\displaystyle A$ is the midpoint of $\displaystyle OX$ and $\displaystyle B$ is the point on $\displaystyle OY$ such that $\displaystyle OB:BY=1:2$. The lines $\displaystyle XB$ and $\displaystyle AY$ intersect at $\displaystyle Z$.

(a) Find $\displaystyle \overrightarrow{AB}$ as a simplified expression in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$.

(b) Using a vector method, find $\displaystyle \overrightarrow{OZ}$ as a simplified expression in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$.

The point $\displaystyle M$ on $\displaystyle XY$ is such that $\displaystyle O$, $\displaystyle Z$ and $\displaystyle M$ are collinear.

(c) Find $\displaystyle \overrightarrow{OM}$ as a simplified expression in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$.

$\displaystyle O$, $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are fixed points such that

(a) Find $\displaystyle \overrightarrow{BC}$ as a simplified expression in terms of $\displaystyle \mathbf{i}$ and $\displaystyle \mathbf{j}$.

(b) Find a unit vector parallel to $\displaystyle \overrightarrow{BC}$.

The point $\displaystyle M$ is the midpoint of $\displaystyle OA$ and the point $\displaystyle N$ lies on $\displaystyle OB$ such that

(c) Show that the points $\displaystyle M$, $\displaystyle N$ and $\displaystyle C$ are collinear.

Referred to a fixed origin $\displaystyle O$, the point $\displaystyle A$ has position vector $\displaystyle (4\mathbf{i}+3\mathbf{j})$ and the point $\displaystyle B$ has position vector $\displaystyle (\mathbf{i}+7\mathbf{j})$.

(a) Find $\displaystyle \overrightarrow{AB}$ as a simplified expression in terms of $\displaystyle \mathbf{i}$ and $\displaystyle \mathbf{j}$.

(b) Find a unit vector that is parallel to $\displaystyle \overrightarrow{AB}$.

Referred (3i (a) Find, respectively. to as a a fixed simplified origin expression the position in terms vectors of i and of the points $\displaystyle P$ and $\displaystyle Q$ are (5i $\displaystyle + 6j)$ and

(b) Find a unit vector parallel to $\displaystyle {\overline{{P Q}}}.$

The Given position that vector of the fixed point $\displaystyle R$ is (13i $\displaystyle +$ aj), where a is a constant.

(c) find the value of a.