Past Exam Questions

Relative to an origin O, the position vectors of the points A and B are given by

\(\overrightarrow{OA} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}\) and \(\overrightarrow{OB} = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}\).

1. Use a scalar product to find angle \(AOB\), correct to the nearest degree.
2. Find the unit vector in the direction of \(\overrightarrow{AB}\).
3. The point C is such that \(\overrightarrow{OC} = 6\mathbf{j} + p\mathbf{k}\), where \(p\) is a constant. Given that the lengths of \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are equal, find the possible values of \(p\).

Relative to an origin O, the position vectors of the points A, B and X are given by

\(\overrightarrow{OA} = \begin{pmatrix} -8 \\ -4 \\ 2 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 10 \\ 2 \\ 11 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OX} = \begin{pmatrix} -2 \\ -2 \\ 5 \end{pmatrix}.\)

(i) Find \(\overrightarrow{AX}\) and show that AXB is a straight line.

The position vector of a point C is given by \(\overrightarrow{OC} = \begin{pmatrix} 1 \\ -8 \\ 3 \end{pmatrix}.\)

(ii) Show that CX is perpendicular to AX.

(iii) Find the area of triangle ABC.

The points A and B have position vectors i + 7j + 2k and -5i + 5j + 6k respectively, relative to an origin O.

(i) Use a scalar product to calculate angle AOB, giving your answer in radians correct to 3 significant figures. [4]

(ii) The point C is such that \(\overrightarrow{AB} = 2\overrightarrow{BC}\). Find the unit vector in the direction of \(\overrightarrow{OC}\). [4]

Relative to an origin O, the position vectors of the points A, B, C and D are given by

\(\overrightarrow{OA} = \begin{pmatrix} 1 \\ 3 \\ -1 \end{pmatrix}, \overrightarrow{OB} = \begin{pmatrix} 3 \\ -1 \\ 3 \end{pmatrix}, \overrightarrow{OC} = \begin{pmatrix} 4 \\ 2 \\ p \end{pmatrix} \text{ and } \overrightarrow{OD} = \begin{pmatrix} -1 \\ 0 \\ q \end{pmatrix}\),

where \(p\) and \(q\) are constants. Find

(i) the unit vector in the direction of \(\overrightarrow{AB}\),

(ii) the value of \(p\) for which angle \(AOC = 90^\circ\),

(iii) the values of \(q\) for which the length of \(\overrightarrow{AD}\) is 7 units.

The points A, B, C and D have position vectors \(3oldsymbol{i} + 2oldsymbol{k}\), \(2oldsymbol{i} - 2oldsymbol{j} + 5oldsymbol{k}\), \(2oldsymbol{j} + 7oldsymbol{k}\) and \(-2oldsymbol{i} + 10oldsymbol{j} + 7oldsymbol{k}\) respectively.

(i) Use a scalar product to show that \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\) are perpendicular. [4]

(ii) Show that \(\overrightarrow{BC}\) and \(\overrightarrow{AD}\) are parallel and find the ratio of the length of \(BC\) to the length of \(AD\). [4]

Given that \(\mathbf{a} = \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}, \mathbf{b} = \begin{pmatrix} 2 \\ 6 \\ 3 \end{pmatrix}\) and \(\mathbf{c} = \begin{pmatrix} p \\ p \\ p+1 \end{pmatrix}\), find

(i) the angle between the directions of \(\mathbf{a}\) and \(\mathbf{b}\),

(ii) the value of \(p\) for which \(\mathbf{b}\) and \(\mathbf{c}\) are perpendicular.

The position vectors of points A and B, relative to an origin O, are given by

\(\overrightarrow{OA} = \begin{pmatrix} 6 \\ -2 \\ -6 \end{pmatrix}\) and \(\overrightarrow{OB} = \begin{pmatrix} 3 \\ k \\ -3 \end{pmatrix}\),

where \(k\) is a constant.

1. Find the value of \(k\) for which angle \(AOB\) is \(90^\circ\).
2. Find the values of \(k\) for which the lengths of \(OA\) and \(OB\) are equal.

The point C is such that \(\overrightarrow{AC} = 2\overrightarrow{CB}\).

3. In the case where \(k = 4\), find the unit vector in the direction of \(\overrightarrow{OC}\).

Two vectors, u and v, are such that

\(\mathbf{u} = \begin{pmatrix} q \\ 2 \\ 6 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} 8 \\ q-1 \\ q^2-7 \end{pmatrix}\),

where \(q\) is a constant.

(i) Find the values of \(q\) for which \(\mathbf{u}\) is perpendicular to \(\mathbf{v}\).

(ii) Find the angle between \(\mathbf{u}\) and \(\mathbf{v}\) when \(q = 0\).

Relative to an origin O, the position vectors of the points A, B and C are given by

\(\overrightarrow{OA} = \begin{pmatrix} 1 \\ -3 \\ 2 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}.\)

1. Find \(\overrightarrow{AC}\).
2. The point M is the mid-point of AC. Find the unit vector in the direction of \(\overrightarrow{OM}\).
3. Evaluate \(\overrightarrow{AB} \cdot \overrightarrow{AC}\) and hence find angle BAC.

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

\(\overrightarrow{OA} = \begin{pmatrix} 8 \\ -6 \\ 5 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} -10 \\ 3 \\ -13 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 2 \\ -3 \\ -1 \end{pmatrix}.\)

A fourth point, \(D\), is such that the magnitudes \(|\overrightarrow{AB}|, |\overrightarrow{BC}|\) and \(|\overrightarrow{CD}|\) are the first, second and third terms respectively of a geometric progression.

(i) Find the magnitudes \(|\overrightarrow{AB}|, |\overrightarrow{BC}|\) and \(|\overrightarrow{CD}|\).

(ii) Given that \(D\) is a point lying on the line through \(B\) and \(C\), find the two possible position vectors of the point \(D\).

In the diagram, OABCDEFG is a cuboid in which OA = 3 units, OC = 2 units and OD = 2 units. Unit vectors i, j and k are parallel to OA, OD and OC respectively. M is the midpoint of EF.

(a) Find the position vector of M.

The position vector of P is i + j + 2k.

(b) Calculate angle PAM.

(c) Find the exact length of the perpendicular from P to the line passing through O and M.

The diagram shows a pyramid OABCD with a horizontal rectangular base OABC. The sides OA and AB have lengths of 8 units and 6 units respectively. The point E on OB is such that OE = 2 units. The point D of the pyramid is 7 units vertically above E. Unit vectors i, j and k are parallel to OA, OC and ED respectively.

(i) Show that \(\overrightarrow{OE} = 1.6\mathbf{i} + 1.2\mathbf{j}\).

(ii) Use a scalar product to find angle BDO.

The diagram shows a three-dimensional shape. The base OAB is a horizontal triangle in which angle AOB is 90°. The side OBCD is a rectangle and the side OAD lies in a vertical plane. Unit vectors i and j are parallel to OA and OB respectively and the unit vector k is vertical. The position vectors of A, B and D are given by \(\overrightarrow{OA} = 8\mathbf{i}\), \(\overrightarrow{OB} = 5\mathbf{j}\) and \(\overrightarrow{OD} = 2\mathbf{i} + 4\mathbf{k}\).

(i) Express each of the vectors \(\overrightarrow{DA}\) and \(\overrightarrow{CA}\) in terms of i, j and k.

(ii) Use a scalar product to find angle CAD.

Fig. 1 shows a rectangle with sides of 7 units and 3 units from which a triangular corner has been removed, leaving a 5-sided polygon OABCD. The sides OA, AB, BC and DO have lengths of 7 units, 3 units, 3 units and 2 units respectively. Fig. 2 shows the polygon OABCD forming the horizontal base of a pyramid in which the point E is 8 units vertically above D. Unit vectors i, j and k are parallel to OA, OD and DE respectively.

(i) Find \(\overrightarrow{CE}\) and the length of \(CE\).

(ii) Use a scalar product to find angle ECA, giving your answer in the form \(\cos^{-1} \left( \frac{m}{\sqrt{n}} \right)\), where m and n are integers.

The diagram shows a trapezium \(OABC\) in which \(OA\) is parallel to \(CB\). The position vectors of \(A\) and \(B\) relative to the origin \(O\) are given by \(\overrightarrow{OA} = \begin{pmatrix} -2 \\ -2 \\ -1 \end{pmatrix}\) and \(\overrightarrow{OB} = \begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}\).

(i) Show that angle \(OAB\) is \(90^\circ\).

The magnitude of \(\overrightarrow{CB}\) is three times the magnitude of \(\overrightarrow{OA}\).

(ii) Find the position vector of \(C\).

(iii) Find the exact area of the trapezium \(OABC\), giving your answer in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers.

The diagram shows a triangular pyramid ABCD. It is given that \(\overrightarrow{AB} = 3\mathbf{i} + \mathbf{j} + \mathbf{k}\), \(\overrightarrow{AC} = \mathbf{i} - 2\mathbf{j} - \mathbf{k}\), and \(\overrightarrow{AD} = \mathbf{i} + 4\mathbf{j} - 7\mathbf{k}\).

(i) Verify, showing all necessary working, that each of the angles \(DAB\), \(DAC\), and \(CAB\) is \(90^\circ\).

(ii) Find the exact value of the area of the triangle \(ABC\), and hence find the exact value of the volume of the pyramid.

[The volume \(V\) of a pyramid of base area \(A\) and vertical height \(h\) is given by \(V = \frac{1}{3}Ah\).]

The diagram shows a cuboid OABCDEFG with a horizontal base OABC in which OA = 4 cm and AB = 15 cm. The height OD of the cuboid is 2 cm. The point X on AB is such that AX = 5 cm and the point P on DG is such that DP = p cm, where p is a constant. Unit vectors i, j and k are parallel to OA, OC and OD respectively.

(i) Express \(\overrightarrow{XP}\) in terms of \(p\), \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\).

(ii) Find the exact value of \(p\) for which angle \(OXP = 60^\circ\).

The diagram shows a pyramid OABC with a horizontal triangular base OAB and vertical height OC. Angles AOB, BOC and AOC are each right angles. Unit vectors i, j and k are parallel to OA, OB and OC respectively, with OA = 4 units, OB = 2.4 units and OC = 3 units. The point P on CA is such that CP = 3 units.

1. Show that \(\overrightarrow{CP} = 2.4\mathbf{i} - 1.8\mathbf{k}\).
2. Express \(\overrightarrow{OP}\) and \(\overrightarrow{BP}\) in terms of \(\mathbf{i}, \mathbf{j}\) and \(\mathbf{k}\).
3. Use a scalar product to find angle BPC.

The diagram shows a cuboid OABCPQRS with a horizontal base OABC in which AB = 6 cm and OA = a cm, where a is a constant. The height OP of the cuboid is 10 cm. The point T on BR is such that BT = 8 cm, and M is the mid-point of AT. Unit vectors i, j and k are parallel to OA, OC and OP respectively.

(i) For the case where a = 2, find the unit vector in the direction of \(\overrightarrow{PM}\).

(ii) For the case where angle \(ATP = \cos^{-1}\left(\frac{2}{7}\right)\), find the value of a.

The diagram shows a pyramid \(OABCX\). The horizontal square base \(OABC\) has side 8 units and the centre of the base is \(D\). The top of the pyramid, \(X\), is vertically above \(D\) and \(XD = 10\) units. The mid-point of \(OX\) is \(M\). The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(\overrightarrow{OA}\) and \(\overrightarrow{OC}\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards.

(i) Express the vectors \(\overrightarrow{AM}\) and \(\overrightarrow{AC}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\).

(ii) Use a scalar product to find angle \(MAC\).