1. 1
   Relative to the origin O, the points P and Q have position vectors $\mathbf{u} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$ and $\mathbf{v} = \mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$ respectively.

   (a)

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

   [1 mark]

   • A$10\mathbf{i} - 7\mathbf{j} - 9\mathbf{k}$
   • B$-10\mathbf{i} + 7\mathbf{j} + 9\mathbf{k}$
   • C$-10\mathbf{i} - 7\mathbf{j} + 9\mathbf{k}$
   • D$2\mathbf{i} - 4\mathbf{j} - 6\mathbf{k}$

   (b)

   Find $(3\mathbf{u})\times\mathbf{v}$.

   [1 mark]

   • A$-90\mathbf{i} + 63\mathbf{j} + 81\mathbf{k}$
   • B$30\mathbf{i} - 21\mathbf{j} - 27\mathbf{k}$
   • C$-10\mathbf{i} + 7\mathbf{j} + 9\mathbf{k}$
   • D$-30\mathbf{i} + 21\mathbf{j} + 27\mathbf{k}$

   (c)

   Find the area of triangle OPQ.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $\mathbf{a}\times\mathbf{b} = 2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}$ and $\mathbf{a}\cdot\mathbf{b} = 12$. Let $\theta$ be the angle between $\mathbf{a}$ and $\mathbf{b}$.

   (a)

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

   [1 mark]

   • A$2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}$
   • B$\mathbf{0}$
   • C$-2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}$
   • D$-12$

   (b)

   Find $(\mathbf{a}+\mathbf{b})\times(\mathbf{a}-\mathbf{b})$.

   [1 mark]

   • A$\mathbf{0}$
   • B$-4\mathbf{i} + 6\mathbf{j} - 12\mathbf{k}$
   • C$4\mathbf{i} - 6\mathbf{j} + 12\mathbf{k}$
   • D$-2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}$

   (c)

   Find the exact value of $\tan\theta$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A triangular shade sail is fixed at the points A$(1, 0, 2)$, B$(5, 4, 4)$ and C$(-2, 4, 3)$. All distances are in metres.

   (a)

   Find $\overrightarrow{AB}\times\overrightarrow{AC}$.

   [3 marks]

   (b)

   Find the area of the sail and the shortest distance from C to the edge AB.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Relative to the origin O, the points P and Q have coordinates $(1, 2, -1)$ and $(3, t, 1)$ respectively, where $t\in\mathbb{R}$. The point R is chosen so that OPRQ is a parallelogram.

   (a)

   (i) Show that $\overrightarrow{OP}\times\overrightarrow{OQ} = (t+2)\mathbf{i} - 4\mathbf{j} + (t-6)\mathbf{k}$.
   (ii) Given that the area of OPRQ is 8, find the possible values of $t$. Give your answers in the form $p + q\sqrt{2}$, where $p, q\in\mathbb{Z}$.

   [6 marks]

   (b)

   (i) Show that $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ are not parallel for any value of $t$.
   (ii) Find the minimum area of parallelogram OPRQ and the value of $t$ for which it occurs.
   (iii) Hence write down the minimum area of triangle OPQ.

   [6 marks]

   Total for question 4: 12 marks

End of questions