The position vectors of the points \(A\) and \(B\), relative to the origin, are

\(\begin{pmatrix}2\\-6\end{pmatrix}\) and \(\begin{pmatrix}-3\\6\end{pmatrix}\), respectively.

(a) Find the displacement vector of \(B\) from \(A\).

(b) Find the distance \(AB\).

(c) The point \(X\) is such that \(3\overrightarrow{AB}=2\overrightarrow{AX}\). Find the position vector of \(X\).

Relative to an origin \(O\), the position vector of point \(P\) is \(3\mathbf{i}-2\mathbf{j}\) and the position vector of point \(Q\) is \(8\mathbf{i}+13\mathbf{j}\).

(a) The point \(R\) is such that \(\overrightarrow{PQ}=5\overrightarrow{PR}\). Find the unit vector in the direction \(\overrightarrow{OR}\).

(b) The position vector of \(S\) relative to \(O\) is \(\lambda\mathbf{j}\). Given that \(RS\) is parallel to \(PQ\), find the value of \(\lambda\).

(a) Find a unit vector in the direction of the vector \(40\mathbf{i}-9\mathbf{j}\).

(b) The position vectors of the points \(P\) and \(Q\) are \(\mathbf{p}\) and \(\mathbf{q}\), respectively. The point \(R\) lies on \(PQ\), between \(P\) and \(Q\), such that \(\dfrac{PR}{PQ}=k\).

(i) Write down the set of possible values of \(k\).

(ii) Given that the position vector of \(R\) is \(\lambda\mathbf{p}+\mu\mathbf{q}\), show that \(\lambda+\mu=1\).

Relative to an origin \(O\), the position vectors of points \(A\), \(B\) and \(C\) are

\(\begin{pmatrix}-5\\-7\end{pmatrix},\qquad \begin{pmatrix}10\\-4\end{pmatrix},\qquad \begin{pmatrix}x\\y\end{pmatrix}\)

respectively.

Given that \(\overrightarrow{AC}=4\overrightarrow{BC}\), find a unit vector in the direction of \(\overrightarrow{OC}\).

The diagram shows triangle \(OAB\), where \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). The point \(P\) lies on \(OA\), where \(OP=\frac34 OA\). The point \(Q\) is the midpoint of \(AB\). The lines \(OB\) and \(PQ\), when extended, meet at \(R\).

(a) Find \(\overrightarrow{AB}\) in terms of \(\mathbf a\) and \(\mathbf b\).

(b) Find \(\overrightarrow{PQ}\) in terms of \(\mathbf a\) and \(\mathbf b\), giving your answer in its simplest form.

It is given that \(n\overrightarrow{PQ}=\overrightarrow{QR}\) and \(\overrightarrow{BR}=k\mathbf b\), where \(n\) and \(k\) are positive constants.

(c) Find \(\overrightarrow{QR}\) in terms of \(n\), \(\mathbf a\) and \(\mathbf b\).

(d) Find \(\overrightarrow{QR}\) in terms of \(k\), \(\mathbf a\) and \(\mathbf b\).

(e) Hence find the values of \(n\) and \(k\).

(a) Find the unit vector in the direction of

\(\begin{pmatrix}5\\-12\end{pmatrix}.\)

(b) Given that

\(\begin{pmatrix}4\\1\end{pmatrix} +k\begin{pmatrix}-2\\3\end{pmatrix} =r\begin{pmatrix}-10\\5\end{pmatrix},\)

find the value of each of the constants \(k\) and \(r\).

(c) Relative to an origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf p\), \(3\mathbf q-\mathbf p\) and \(9\mathbf q-5\mathbf p\) respectively.

(i) Find \(\overrightarrow{AB}\) in terms of \(\mathbf p\) and \(\mathbf q\).

(ii) Find \(\overrightarrow{AC}\) in terms of \(\mathbf p\) and \(\mathbf q\).

(iii) Explain why \(A\), \(B\) and \(C\) all lie in a straight line.

(iv) Find the ratio \(AB:BC\).

Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are \(2\mathbf i+12\mathbf j\) and \(6\mathbf i-4\mathbf j\) respectively.

(i) Write down and simplify an expression for \(\overrightarrow{AB}\).

The point \(C\) lies on \(\overrightarrow{AB}\) such that \(AC:CB\) is \(1:3\).

(ii) Find the unit vector in the direction of \(\overrightarrow{OC}\).

The point \(D\) lies on \(\overrightarrow{OA}\) such that \(OD:DA\) is \(1:3\).

(iii) Find an expression for \(\overrightarrow{AD}\) in terms of \(\mathbf i\) and \(\mathbf j\).

(a) Vectors \(\mathbf a\), \(\mathbf b\) and \(\mathbf c\) are such that

\(\mathbf a=\binom{5}{-6},\qquad \mathbf b=\binom{11}{-15},\qquad 3\mathbf a+\mathbf c=\mathbf b.\)

(i) Find \(\mathbf c\).

(ii) Find the unit vector in the direction of \(\mathbf b\).

(b) In the diagram, \(\overrightarrow{OP}=\mathbf p\) and \(\overrightarrow{OQ}=\mathbf q\). The point \(R\) lies on \(PQ\) such that \(PR=3RQ\).

Find \(\overrightarrow{OR}\) in terms of \(\mathbf p\) and \(\mathbf q\), simplifying your answer.