Past Exam Questions

The point P has coordinates (-1, 4, 11) and the line l has equation \(\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ -4 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}\).

Find the perpendicular distance from P to l.

With respect to the origin O, the position vectors of two points A and B are given by \(\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\) and \(\overrightarrow{OB} = 3\mathbf{i} + 4\mathbf{j}\). The point P lies on the line through A and B, and \(\overrightarrow{AP} = \lambda \overrightarrow{AB}\).

1. Show that \(\overrightarrow{OP} = (1 + 2\lambda)\mathbf{i} + (2 + 2\lambda)\mathbf{j} + (2 - 2\lambda)\mathbf{k}\).
2. By equating expressions for \(\cos AOP\) and \(\cos BOP\) in terms of \(\lambda\), find the value of \(\lambda\) for which \(OP\) bisects the angle \(AOB\).
3. When \(\lambda\) has this value, verify that \(AP : PB = OA : OB\).

With respect to the origin O, the lines l and m have vector equations r = 2i + k + \(\lambda\)(i - j + 2k) and r = 2j + 6k + \(\mu\)(i + 2j - 2k) respectively.

1. Prove that l and m do not intersect.
2. Calculate the acute angle between the directions of l and m.

With respect to the origin O, the points A and B have position vectors given by \(\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\) and \(\overrightarrow{OB} = 3\mathbf{i} + 4\mathbf{j}\). The point P lies on the line AB and OP is perpendicular to AB.

(i) Find a vector equation for the line AB.

(ii) Find the position vector of P.

The lines l and m have vector equations

\(\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + s(\mathbf{i} - \mathbf{j} + 2\mathbf{k})\)

and

\(\mathbf{r} = 4\mathbf{i} + 6\mathbf{j} + \mathbf{k} + t(2\mathbf{i} + 2\mathbf{j} + \mathbf{k})\)

respectively.

1. Show that l and m intersect.
2. Calculate the acute angle between the lines.

With respect to the origin O, the points A, B and C have position vectors given by \(\overrightarrow{OA} = \mathbf{i} - \mathbf{k}\), \(\overrightarrow{OB} = 3\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}\) and \(\overrightarrow{OC} = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}\).

The mid-point of AB is M. The point N lies on AC between A and C and is such that \(AN = 2NC\).

(i) Find a vector equation of the line MN.

(ii) It is given that MN intersects BC at the point P. Find the position vector of P.

The points A and B have position vectors, relative to the origin O, given by \(\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) and \(\overrightarrow{OB} = 2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\).

The line \(l\) has vector equation \(\mathbf{r} = (1 - 2t)\mathbf{i} + (5 + t)\mathbf{j} + (2 - t)\mathbf{k}\).

(i) Show that \(l\) does not intersect the line passing through A and B.

(ii) The point P lies on \(l\) and is such that angle \(PAB\) is equal to 60°. Given that the position vector of P is \((1 - 2t)\mathbf{i} + (5 + t)\mathbf{j} + (2 - t)\mathbf{k}\), show that \(3t^2 + 7t + 2 = 0\). Hence find the only possible position vector of P.

The points A and B have position vectors, relative to the origin O, given by

\(\overrightarrow{OA} = \begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix}\) and \(\overrightarrow{OB} = \begin{pmatrix} 3 \\ -1 \\ -4 \end{pmatrix}\).

The line l passes through A and is parallel to OB. The point N is the foot of the perpendicular from B to l.

(i) State a vector equation for the line l.

(ii) Find the position vector of N and show that \(BN = 3\).

With respect to the origin O, the points A and B have position vectors given by \(\overrightarrow{OA} = 2\mathbf{i} + 2\mathbf{j} + \mathbf{k}\) and \(\overrightarrow{OB} = \mathbf{i} + 4\mathbf{j} + 3\mathbf{k}\).

The line l has vector equation \(\mathbf{r} = 4\mathbf{i} - 2\mathbf{j} + 2\mathbf{k} + s(\mathbf{i} + 2\mathbf{j} + \mathbf{k})\).

Prove that the line l does not intersect the line through A and B.

With respect to the origin O, the points A, B, C and D have position vectors given by

\(\overrightarrow{OA} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \quad \overrightarrow{OC} = \begin{pmatrix} 1 \\ -2 \\ 5 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OD} = \begin{pmatrix} 5 \\ -6 \\ 11 \end{pmatrix}.\)

(a) Find the obtuse angle between the vectors \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\).

The line \(l\) passes through the points \(A\) and \(B\).

(b) Find a vector equation for the line \(l\).

(c) Find the position vector of the point of intersection of the line \(l\) and the line passing through \(C\) and \(D\).

The lines l and m have vector equations

\(\mathbf{r} = 2\mathbf{i} - \mathbf{j} + 4\mathbf{k} + s(\mathbf{i} + \mathbf{j} - \mathbf{k})\)

and

\(\mathbf{r} = -2\mathbf{i} + 2\mathbf{j} + \mathbf{k} + t(-2\mathbf{i} + \mathbf{j} + \mathbf{k})\)

respectively.

1. Show that l and m do not intersect.
2. The point P lies on l and the point Q has position vector \(2\mathbf{i} - \mathbf{k}\). Given that the line PQ is perpendicular to l, find the position vector of P.
3. Verify that Q lies on m and that PQ is perpendicular to m.

The lines l and m have vector equations

\(\mathbf{r} = \mathbf{i} - 2\mathbf{k} + s(2\mathbf{i} + \mathbf{j} + 3\mathbf{k})\)

and

\(\mathbf{r} = 6\mathbf{i} - 5\mathbf{j} + 4\mathbf{k} + t(\mathbf{i} - 2\mathbf{j} + \mathbf{k})\)

respectively.

Show that l and m intersect, and find the position vector of their point of intersection.

With respect to the origin O, the points A, B, C, D have position vectors given by

\(\overrightarrow{OA} = 4\mathbf{i} + \mathbf{k}, \quad \overrightarrow{OB} = 5\mathbf{i} - 2\mathbf{j} - 2\mathbf{k}, \quad \overrightarrow{OC} = \mathbf{i} + \mathbf{j}, \quad \overrightarrow{OD} = -\mathbf{i} - 4\mathbf{k}\)

1. Calculate the acute angle between the lines AB and CD.
2. Prove that the lines AB and CD intersect.
3. The point P has position vector \(\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\). Show that the perpendicular distance from P to the line AB is equal to \(\sqrt{3}\).

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

\(\overrightarrow{OA} = \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 4 \\ -3 \\ -2 \end{pmatrix}.\)

The midpoint of AC is M and the point N lies on BC, between B and C, and is such that BN = 2NC.

(a) Find the position vectors of M and N.

(b) Find a vector equation for the line through M and N.

(c) Find the position vector of the point Q where the line through M and N intersects the line through A and B.

With respect to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow{OA} = \mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\). The line \(l\) has vector equation \(\mathbf{r} = 4\mathbf{i} + \mathbf{k} + \lambda (-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\).

(a) Find in degrees the acute angle between the directions of \(OA\) and \(l\).

(b) Find the position vector of the foot of the perpendicular from \(A\) to \(l\).

(c) Hence find the position vector of the reflection of \(A\) in \(l\).

The lines l and m have vector equations

\(\mathbf{r} = -\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} - \mathbf{k})\)

and

\(\mathbf{r} = 5\mathbf{i} + 4\mathbf{j} + 3\mathbf{k} + \mu(a\mathbf{i} + b\mathbf{j} + \mathbf{k})\)

respectively, where a and b are constants.

(a) Given that l and m intersect, show that \(2b - a = 4\).

(b) Given also that l and m are perpendicular, find the values of a and b.

(c) When a and b have these values, find the position vector of the point of intersection of l and m.

In the diagram, OABCDEFG is a cuboid in which OA = 2 units, OC = 4 units and OG = 2 units. Unit vectors i, j and k are parallel to OA, OC and OG respectively. The point M is the midpoint of DF. The point N on AB is such that AN = 3NB.

(a) Express the vectors \(\overrightarrow{OM}\) and \(\overrightarrow{MN}\) in terms of i, j and k.

(b) Find a vector equation for the line through M and N.

(c) Show that the length of the perpendicular from O to the line through M and N is \(\sqrt{\frac{53}{6}}\).

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

\(\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 4 \\ 3 \\ 2 \end{pmatrix}, \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 3 \\ -2 \\ -4 \end{pmatrix}.\)

The quadrilateral \(ABCD\) is a parallelogram.

(a) Find the position vector of \(D\).

(b) The angle between \(BA\) and \(BC\) is \(\theta\). Find the exact value of \(\cos \theta\).

(c) Hence find the area of \(ABCD\), giving your answer in the form \(p\sqrt{q}\), where \(p\) and \(q\) are integers.

(a) Relative to an origin O, the position vectors of two points P and Q are p and q respectively. The point R is such that PQR is a straight line with Q the mid-point of PR. Find the position vector of R in terms of p and q, simplifying your answer.

(b) The vector 6i + aj + bk has magnitude 21 and is perpendicular to 3i + 2j + 2k. Find the possible values of a and b, showing all necessary working.

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

\(\overrightarrow{OA} = \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix}\) and \(\overrightarrow{OB} = \begin{pmatrix} 5 \\ 4 \\ -3 \end{pmatrix}\).

The point P lies on AB and is such that \(\overrightarrow{AP} = \frac{1}{3} \overrightarrow{AB}\).

(i) Find the position vector of P.

(ii) Find the distance OP.

(iii) Determine whether OP is perpendicular to AB. Justify your answer.