Exam-Style Questions.

1.	GCSE Higher

(a) Shape \(A\) is translated to shape \(B\) using the vector \( \begin{pmatrix}m\\n\\ \end{pmatrix}\). What are the values of \(m\) and \(n\)?

(b) Vectors \(a, b, c, d\) and \(e\) are drawn on an isometric grid. Write each of the vectors \(c, d\) and \(e\) in terms of \(a\) and/or \(b\).

Worked Solution

---

2.	IB Standard

Consider two perpendicular vectors \(p\) and \(q\).

(a) Let \(r=p-q\). Draw a diagram to show what this relationship might look like.

(b) If \(p=\begin{pmatrix} 4 \\ 1 \\ -3 \\ \end{pmatrix}\) and \(q=\begin{pmatrix} 3 \\ n \\ -5 \\ \end{pmatrix}\), where \(n\in \mathbb Z\), find \(n\).

Worked Solution

---

3.	IB Standard

George and Hugo like to fly model airplanes. On one day George's plane takes off from level ground and shortly after that Hugo's plane takes off.

The position of George’s plane \(s\) seconds after it takes off is given by \(\begin{pmatrix} 1 \\ 2 \\ 0 \\ \end{pmatrix} + s\begin{pmatrix} 5 \\ -2 \\ 6 \\ \end{pmatrix} \) where the distances are in metres.

(a) Find the speed of George’s plane to the nearest integer.

(b) Find the height of George’s plane after four seconds.

The position of Hugo’s airplane \(t\) seconds after it takes off is given by \(\begin{pmatrix} 4 \\ -4 \\ 0 \\ \end{pmatrix}+t\begin{pmatrix} 7 \\ -2 \\ 9 \\ \end{pmatrix} \) where the distances are in metres.

(c) Show that the paths of the planes are not perpendicular.

The two airplanes collide at the point \((46, -16, 54)\).

(d) How long after George’s plane takes off does Hugo’s plane take off ?

Worked Solution

---

4.	IB Standard

The line \(L_1\) passes through the points A(3, 5, 1) and B(3, 6, 0).

(a) Show that \(AB=\begin{pmatrix} 0 \\ 1 \\ -1 \\ \end{pmatrix}\)

(b) Find a direction vector for \(L_1\)

(c) a vector equation for \(L_1\)

Another line \(L_2\) has equation \(\begin{pmatrix} 8 \\ 3 \\ -2 \\ \end{pmatrix} + t \begin{pmatrix} -1 \\ 1 \\ 0 \\ \end{pmatrix}\). The lines \(L_1\) and \(L_2\) intersect at the point P.

(d) Find the coordinates of P.

(e) Find a direction vector for \(L_2\).

(f) Hence, find the angle between \(L_1\) and \(L_2\).

Worked Solution

---

5.	IB Standard

Two points \(A\) and \(B\) have coordinates (1 , 3 , 6) and (8 , 7 , 10) respectively.

(a) Find \( \overrightarrow{AB} \) in terms of the unit vectors \(i, j\) and \(k\).

(b) Find \(\mid\overrightarrow{AB} \mid\)

Let \( \overrightarrow{AC} = 5i + 2j - k\)

(c) Find the angle between \(AB\) and \(AC\).

(d) Find the area of triangle \(ABC\).

(e) Hence or otherwise find the shortest distance from \(C\) to the line through \(A\) and \(B\).

Worked Solution

---

6.

A-Level

The points A and B have coordinates \((3,-2,1)\) and \((4, 0, -1)\) respectively.

The line \(l\) has the following equation:

$$ r= \begin{pmatrix} 3 \\ 2 \\ 0 \\ \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ -2 \\ 0 \\ \end{pmatrix} $$

The point C lies on \(l\) where \(\lambda = 3\).

(a) Find the coordinates of C.

(b) Find the acute angle ABC, giving your answer to the nearest tenth of a degree.

(c) The point D lies on a line through A and B such that angle ADC is a right angle. Find the coordinates of D.

(d) The point E completes the parallelogram ACBE. Find the coordinates of E.

Worked Solution

---

7.	IB Analysis and Approaches

Consider the vectors \(\mathbf{a}\) and \(\mathbf{b}\) such that \(\mathbf{a} = \begin{pmatrix} 16 \\ -12 \end{pmatrix} \) and \( |\mathbf{b}| = 11\).

(a) Find the possible range of values for \(|\mathbf{a+b}|\).

Consider the vector \(\mathbf{p}\) such that \(\mathbf{p=a+b}\).

(b) Given that \(|\mathbf{a+b}|\) is a minimum, find \(\mathbf{p}\).

Consider the vector q such that \(\mathbf{q} = \begin{pmatrix}x \\ y \end{pmatrix} \) , where \(x,y \in \mathbb{R} \).

(c) Find \(\mathbf{q}\) such that \(\mathbf{|q| = |b|}\) and \(\mathbf{q}\) is perpendicular to \(\mathbf{a}\).

Worked Solution

---