(a) Given that \(\overrightarrow{PQ}=\binom{-3}{7}\) and \(4\overrightarrow{PR}=\binom{-2}{8}\), find \(\overrightarrow{RQ}\).

(b) The vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are such that \(\mathbf{a}=\alpha\mathbf{i}+6\mathbf{j}\), \(\mathbf{b}=4\mathbf{i}+\beta\mathbf{j}\) and \(\mathbf{c}=(2\alpha+5\beta)\mathbf{i}+20\mathbf{j}\), where \(\alpha\) and \(\beta\) are scalars.

Given that \(\mathbf{c}=3\mathbf{a}-2\mathbf{b}\), find the values of \(\alpha\) and \(\beta\).

(a) The position vectors of the points \(P\), \(Q\) and \(R\), relative to an origin \(O\), are

\(\begin{pmatrix}4\\7\end{pmatrix},\quad \begin{pmatrix}8\\5\end{pmatrix},\quad \begin{pmatrix}x\\y\end{pmatrix}\)

respectively. The point \(R\) lies on \(PQ\) extended such that \(3\overrightarrow{QR}=2\overrightarrow{PR}\). Use a vector method to find the values of \(x\) and \(y\).

(b) You are given that \(\mathbf i\) is a unit vector due east and \(\mathbf j\) is a unit vector due north.

Three vectors, \(\mathbf a\), \(\mathbf b\) and \(\mathbf c\), are in the same horizontal plane as \(\mathbf i\) and \(\mathbf j\), and are such that \(\mathbf a+\mathbf b=\mathbf c\). The magnitude and bearing of \(\mathbf a\) are \(5\) and \(210^\circ\). The magnitude and bearing of \(\mathbf c\) are \(10\) and \(330^\circ\).

(i) Find \(\mathbf a\) and \(\mathbf c\) in terms of \(\mathbf i\) and \(\mathbf j\).

(ii) Find the magnitude and bearing of \(\mathbf b\).

(a) Find the vector which is in the opposite direction to \(\begin{pmatrix}15\\-8\end{pmatrix}\) and has magnitude \(8.5\).

(b) Find \(a\) and \(b\) such that

\(5\begin{pmatrix}3a\\b\end{pmatrix}+\begin{pmatrix}2a+1\\2\end{pmatrix}=6\begin{pmatrix}b+a\\2\end{pmatrix}.\)

(a) Find the unit vector in the same direction as \(\begin{pmatrix}-15\\8\end{pmatrix}\).

(b) Given that

\(\begin{pmatrix}2a\\-5\end{pmatrix}+\begin{pmatrix}4b-12\\3\end{pmatrix}=4\begin{pmatrix}b-a\\a+2b\end{pmatrix},\)

find the values of \(a\) and \(b\).

(a) In this question, \(\mathbf{i}\) is a unit vector due east and \(\mathbf{j}\) is a unit vector due north. A cyclist rides at a speed of 4 m s\(^{-1}\) on a bearing of \(015^{\circ}\). Write the velocity vector of the cyclist in the form \(x\mathbf{i}+y\mathbf{j}\), where \(x\) and \(y\) are constants.

(b) A vector of magnitude 6 on a bearing of \(300^{\circ}\) is added to a vector of magnitude 2 on a bearing of \(230^{\circ}\) to give a vector \(\mathbf{v}\). Find the magnitude and bearing of \(\mathbf{v}\).

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

\(\overrightarrow{OA}=\binom{6}{-5},\quad \overrightarrow{OB}=\binom{10}{3},\quad \overrightarrow{OC}=\binom{x}{y},\quad \overrightarrow{OD}=\binom{12}{7}.\)

(a) Find the unit vector in the direction of \(\overrightarrow{AB}\).

(b) The point \(A\) is the mid-point of \(BC\). Find the value of \(x\) and of \(y\).

(c) The point \(E\) lies on \(OD\) such that \(OE:OD=1:1+\lambda\). Find the value of \(\lambda\) such that \(\overrightarrow{BE}\) is parallel to the \(x\)-axis.

(a) Find the vector which has magnitude \(39\) and is in the same direction as \(\begin{pmatrix}12\\-5\end{pmatrix}\).

(b) Given that \(\mathbf a=\begin{pmatrix}2\\-1\end{pmatrix}\) and \(\mathbf b=\begin{pmatrix}-4\\5\end{pmatrix}\), find the constants \(\lambda\) and \(\mu\) such that \(5\mathbf a+\lambda\begin{pmatrix}4\\6\end{pmatrix}=\mu\mathbf b\).

The vector \(\mathbf p\) has magnitude \(39\) and is in the direction \(-5\mathbf i+12\mathbf j\). The vector \(\mathbf q\) has magnitude \(34\) and is in the direction \(15\mathbf i-8\mathbf j\).

(a) Write both \(\mathbf p\) and \(\mathbf q\) in terms of \(\mathbf i\) and \(\mathbf j\).

(b) Find the magnitude of \(\mathbf p+\mathbf q\) and the angle this vector makes with the positive \(x\)-axis.

The vectors \(\mathbf a\) and \(\mathbf b\) are such that

\(\mathbf a=\alpha\mathbf i+\mathbf j \qquad\text{and}\qquad \mathbf b=12\mathbf i+\beta\mathbf j.\)

(a) Find the value of each of the constants \(\alpha\) and \(\beta\) such that

\(4\mathbf a-\mathbf b=(\alpha+3)\mathbf i-2\mathbf j.\)

(b) Hence find the unit vector in the direction of \(\mathbf b-4\mathbf a\).

Vectors \(\mathbf{i}\) and \(\mathbf{j}\) are vectors parallel to the \(x\)-axis and \(y\)-axis respectively.

Given that

\(\mathbf{a}=2\mathbf{i}+3\mathbf{j},\qquad \mathbf{b}=\mathbf{i}-5\mathbf{j},\qquad \mathbf{c}=3\mathbf{i}+11\mathbf{j},\)

find

(i) the exact value of \(|\mathbf{a}+\mathbf{c}|\),

(ii) the value of \(m\) such that \(\mathbf{a}+m\mathbf{b}\) is parallel to \(\mathbf{j}\),

(iii) the value of \(n\) such that \(n\mathbf{a}-\mathbf{b}=\mathbf{c}\).

(a) The vector \(\mathbf v\) has a magnitude of 39 units and is in the same direction as \(\begin{pmatrix}-12\\5\end{pmatrix}\). Write \(\mathbf v\) in the form \(\begin{pmatrix}a\\b\end{pmatrix}\), where \(a\) and \(b\) are constants.

(b) Vectors \(\mathbf p\) and \(\mathbf q\) are such that \(\mathbf p=\begin{pmatrix}r+s\\r+6\end{pmatrix}\) and \(\mathbf q=\begin{pmatrix}5r+1\\2s-1\end{pmatrix}\), where \(r\) and \(s\) are constants. Given that \(2\mathbf p+3\mathbf q=\begin{pmatrix}0\\0\end{pmatrix}\), find the value of \(r\) and of \(s\).

Vectors \(\mathbf i\) and \(\mathbf j\) are unit vectors parallel to the \(x\)-axis and \(y\)-axis respectively.

(a) The vector \(\mathbf v\) has magnitude \(3\sqrt5\) and has the same direction as \(\mathbf i-2\mathbf j\). Find \(\mathbf v\) in the form \(a\mathbf i+b\mathbf j\), where \(a\) and \(b\) are integers.

(b) The velocity vector \(\mathbf w\) makes an angle of \(30^\circ\) with the positive \(x\)-axis and \(|\mathbf w|=2\). Find \(\mathbf w\) in the form \(\sqrt c\,\mathbf i+d\mathbf j\), where \(c\) and \(d\) are integers.