1. 1
   The vectors $\mathbf{u}$ and $\mathbf{v}$ are given by $\mathbf{u}=2\mathbf{i}-\mathbf{j}+3\mathbf{k}$ and $\mathbf{v}=\mathbf{i}+4\mathbf{j}+2\mathbf{k}$.

   (a)

   Find $\mathbf{u}\cdot\mathbf{v}$.

   [1 mark]

   • A$4$
   • B$12$
   • C$2\mathbf{i}-4\mathbf{j}+6\mathbf{k}$
   • D$\sqrt{294}$

   (b)

   Find $\mathbf{u}\cdot(\mathbf{u}+\mathbf{v})$.

   [1 mark]

   • A$35$
   • B$14$
   • C$4$
   • D$18$

   (c)

   Find the angle between $\mathbf{u}$ and $\mathbf{v}$. Give your answer in degrees, correct to one decimal place. A calculator may be used.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $|\mathbf{a}|=3$ and $|\mathbf{b}|=4$, and the angle between them is $60^\circ$.

   (a)

   Find $\mathbf{a}\cdot\mathbf{b}$.

   [1 mark]

   • A$12$
   • B$6$
   • C$6\sqrt3$
   • D$7$

   (b)

   Find $|\mathbf{a}+\mathbf{b}|$.

   [1 mark]

   • A$7$
   • B$5$
   • C$\sqrt{37}$
   • D$\sqrt{13}$

   (c)

   Find the value of $k$ for which $\mathbf{a}+k\mathbf{b}$ is perpendicular to $\mathbf{a}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The points $A$, $B$ and $C$ have coordinates $A(1,0,2)$, $B(3,2,1)$ and $C(0,t,4)$, where $t\in\mathbb{R}$. The angle $B\hat{A}C$ is a right angle.

   (a)

   Find the value of $t$.

   [3 marks]

   (b)

   Show that triangle $ABC$ is isosceles and find the exact value of $\cos A\hat{B}C$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A crane moves a crate in a straight line. The constant force exerted on the crate by the crane's cable is $\mathbf{F}=(2\mathbf{i}+3\mathbf{j}+6\mathbf{k})$ kN and the displacement of the crate is $\mathbf{d}=(4\mathbf{i}-2\mathbf{j}+4\mathbf{k})$ m. The work done by a constant force $\mathbf{F}$ over a displacement $\mathbf{d}$ is $W=\mathbf{F}\cdot\mathbf{d}$; with force in kN and distance in m, $W$ is in kJ.

   (a)

   (i) Find the work done by the force, stating the units.
   (ii) Find the angle between $\mathbf{F}$ and $\mathbf{d}$, in degrees correct to one decimal place. A calculator may be used.
   (iii) The same force acts on a second crate, whose displacement is $(\mathbf{i}+2\mathbf{j}+c\mathbf{k})$ m. The force does no work on this crate. Find the value of $c$.

   [6 marks]

   (b)

   The force can be split into a part along the direction of motion and a part perpendicular to it.
   (i) Find the value of $\lambda$ for which $\mathbf{F}-\lambda\mathbf{d}$ is perpendicular to $\mathbf{d}$.
   (ii) Let $\mathbf{w}=\mathbf{F}-\lambda\mathbf{d}$, with $\lambda$ as in (i). Using properties of the scalar product, show that $|\mathbf{F}|^2=|\lambda\mathbf{d}|^2+|\mathbf{w}|^2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions