Mathematics Workbook

Question Grading

The questions will be graded as follows:

1. ♥ - Core topics covered at A-Level. These should be familiar to all, and lectures will normally assume a strong knowledge of these.

2. ♣ - A somewhat harder question from the Core topics.

3. ♠ - Topics not covered in all core modules; may include Further Maths. These will be covered in lectures, however, it is worth getting to grips with the content before, as content may be covered at a fairly fast pace.

1.1 Sketch the graphs of the following functions ♥

1. \(f_1(x) = \ln(x)\)
2. \(f_2(x) = \arcsin(x)\)
3. \(f_3(x) = \frac{1}{x}\)
4. \(f_4(x) = |3-x|\)
5. \(f_5(x) = x-|x|\)
6. \(f_6(x) = \sin(x+\pi)\)
7. \(f_7(x) = \cos(2x+1)\)
8. \(f_8(x) = x^2 - 2\)
9. \(f_9(x) = x^3\)
10. \(f_{10}(x) = e^{-x^2}\)
11. \(f_{11}(x) = |4-x^2|\)
12. \(f_{12}(x) = \tan(x)\)

Note: You will be expected to be able to draw the graphs of elementary functions, and to perform basic manipulations of graphs (horizontal and vertical translation, symmetry w.r.t. the y-axis, the x-axis and the origin, dilatations and contractions).

2.1 Factorization ♥

Factorise the following polynomial expressions:

1. \(x^2 - 3x + 2\)
2. \(3x^3 - 3x^2 - 6x\)
3. \(x^2 - x - 1\)
4. \(x^3 - 1\)
5. \(x^4 - 3x^3 - 3x^2 + 11x - 6\)

Note: In part (c) you will need the quadratic formula to find the factors; part (d) has one linear and one quadratic factor; for part (e) you can use the factor theorem.

2.2 More factorization ♥

Find the values of \(x\) for which \(x^3 < 2x^2 + 3x\).

2.3 Partial fractions ♥

Partial fractions provide a powerful method that can reduce complicated looking integrals into trivial ones. It has other uses besides this, however this is most common one that you will encounter.

Express the following as partial fractions:

1. \(\frac{2}{(x+1)(x-1)}\)
2. \(\frac{1}{x^3 + 1}\)
3. \(\frac{4x+1}{(x+1)^2(x-2)}\)
4. \(\frac{x^2-7}{(x-2)(x+1)}\)

Note: It is best not to use the ‘cover-up rule’; there are at least two other ways which use elementary mathematics, whereas the cover-up rule works for more sophisticated reasons. To most, it simply acts as a recipe (which does not always work).

2.4 Completing the square ♥

Find the smallest value for (for real \(x\) and \(y\)):

1. \(x^2 - 2x + 6\)
2. \(x^4 + 2x^2 + y^4 - 2y^2 + 3\)
3. \(\sin^2(x) + 4\sin(x)\)

2.5 Binomial Expansions ♣

A remarkably useful tool, particularly in making approximations up to a certain order.

1. State the generalised binomial theorem for:

   1. \((x+y)^n\)
   2. \((1+x)^n\)
   3. \((1+X)^n\) when \(X = X(x)\), i.e \(X\) is any function of \(x\)

2. Find the coefficient of \(x^k\) for \(0 \leq k \leq 10\) in the expansion of \((2+3x)^{10}\)

3. Find the expansion in powers of \(x\) up to \(x^4\) of \((1+x+x^2)^6\)

4. Find the expansion in powers of \(x\) up to \(x^4\) of \((1-x^3)^6(1-x)^6\)

5. Find the first four terms in the expansion of \(\sqrt{2+x}\)

2.6 Taylor Series and Taylor Expansions ♠

Taylor series are incredibly useful, both in physics and mathematics. They represent an infinitely differentiable function as an infinite series of terms, relating to derivatives of the function at the particular point. This is known as Taylor expanding about a point. Choosing the point \(x=0\) is known as the Maclaurin series, and is the most common special case of general Taylor series.

Taylor series are not on all core syllabuses, however, are worth studying before arriving in Durham. Many of you who studied Further Maths will have encountered them before; regardless, it is still worth going over them again. Many proofs and derivations will use Taylor series as an approximation or representation to derive general results.

We define a Taylor series about the point \(a\) as:

\[f(x) \approx f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots\]

\[= \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\]

By substituting our function into this definition, we can find a nice approximation of the function. The more terms in the function, the more accurate the approximation.

Tasks:

1. By substituting the appropriate function into the definition about the point \(a=0\), derive the following series expansions:

   1. \(\sin(x) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1}\)

   2. \(\cos(x) \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n}\)

   3. \(\frac{1}{1-x} \approx 1 + x + x^2 + x^3 + \ldots = \sum_{n=0}^{\infty} x^n\)

   4. \(e^x \approx 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots = \sum_{n=0}^{\infty} \frac{x^n}{n!}\)

   Note: These are only some of the more common functions; others useful expansions would include logarithms and tan. For completeness, one should derive series for these.

2. Recall the binomial expansion of the function \((1+x)^n\). How can we express this as a Taylor series? What do we find when we take \(n=-1\)? How is this related to part iii) above?

3.1 Small angle approximations ♠

Small angle approximations are valid and useful in a surprisingly large range of trigonometric circumstances, from mechanics to optics to differential equations to astronomy to quantum mechanics.

We consider the effect of each term as \(x\) (or \(\theta\)) approaches zero. For example, \(x^3\) will decrease much more quickly than \(x\) as \(x\) tends to zero, and so we can safely neglect the \(x^3\) term. Hence in the small angle limit, the function \(f(x) = x + x^3 \approx x\).

Using this idea, and the Taylor series above, find the small angle approximation of:

1. \(\sin(x)\)
2. \(\cos(x)\)
3. \(\tan(x)\)

3.2 Trigonometric Equations ♥

1. Solve the following trig equations ♥:
   1. \(\sin(x+\frac{\pi}{6}) + \sin(x-\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\)
   2. \(\cos(x) + \cos(2x) + \cos(3x) = 0\)
2. Solve the following problem (harder) ♠:
   1. Write down the value of \(\cot(\frac{\pi}{6})\)
   2. Use a double angle formula to show that \(\cot(\frac{\pi}{12})\) satisfies the equation \(c^2 - 2\sqrt{3}c - 1 = 0\)
   3. Deduce that \(\cot(\frac{\pi}{12}) = 2 + \sqrt{3}\)

3.3 A simple harmonic pendulum - more involved question ♣

Figure: Simple pendulum. \(h\) represents the maximum height risen by the bob

This question will come in two parts, which are not related.

1. By analysing the geometry, prove that the height of the bob in the small angle approximation, is given by \(h = \frac{1}{2}L\theta_{max}^2\).

2. From the rotational form of Newton’s Second Law, it is given that \(ml^2\ddot{\theta} = -mgl\sin(\theta)\). Using this result, prove that the pendulum obeys simple harmonic motion in the small angle limit.

Hint: In (a), identify what values we need to know to find an expression for the height. Consider a right-angled triangle to help find this needed value in terms of L. For (b), consider the definition of SHM, and the general form of a differential equation governing SHM.

4.1 Direct Differentiation ♥

Differentiate \(y(x)\) with respect to \(x\):

1. \(y = (5x+2)^4\)
2. \(y = \sin(3x+2)\)
3. \(y = e^{4x-1}\)
4. \(y = 5\cos(2x+3)\)
5. \(y = \cos^3(x)\)
6. \(y = \ln(4x-5)\)

4.2 Chain and Product Rule ♣

Find the derivatives of the functions, where \(\alpha\) and \(\beta\) are constants:

1. \(y = \alpha\sin(x^2)\)
2. \(y = \ln(x^\beta + x^{-\beta})\)

4.3 Stationary Points and Graph Sketching ♥

Find the stationary points of the following function:

\[f(x) = \frac{x}{x^2 + a^2}\]

where \(a > 0\), classifying them as maximum or minimum. Sketch the curve.

5.1 Indefinite Integrals ♥/♣

Calculate the following integrals ♥:

1. \(\int x^4 dx\)
2. \(\int 3\sin(x) dx\)
3. \(\int 4e^x dx\)
4. \(\int a^x dx\)
5. \(\int x^{-4} dx\)
6. \(\int 2\sec^2(x) dx\)

Calculate the following integrals ♣:

1. \(\int \frac{1}{x^2 - 2x + 6} dx\)
2. \(\int \frac{1}{1-x^3} dx\)
3. \(\int \arctan(x) dx\)
4. \(\int x^3 e^{x^2} dx\)

5.2 Integrals of even and odd functions ♠

Many of you will have encountered even and odd functions, but perhaps did not appreciate how useful they can be.

• An odd function is defined as a function such that \(f(x) = -f(-x)\). In other words, the function is rotationally symmetric about the origin.
• An even function is defined as a function such that \(f(x) = f(-x)\). In other words, the function is symmetric about the \(y\)-axis.

In A-Levels you may have seen these in terms of trigonometric functions. An important property of the integrals of such functions is that:

• The integral of an odd function over a symmetric interval is 0: \[\int_{-a}^{a} f_{odd}(x) dx = 0\]

• The integral of an even function over a symmetric interval is twice the integral over half of the domain: \[\int_{-a}^{a} f_{even}(x) dx = 2\int_{0}^{a} f_{even}(x) dx\]

Calculate the following integrals, using parity properties of functions ♠:

1. \(\int_{-5}^{5} x^3 dx\)
2. \(\int_{-a}^{a} 94x^7 + x^3 dx\)
3. \(\int_{-\infty}^{\infty} \left(\frac{\pi}{\sqrt{e}}\right)^\pi e^{-\pi x^2} x^3(\sin(x)\cos(x)\tan(x))^3 dx\)

Note: The product of an even function and an odd function is odd.

Figure: Red: Even functions, Blue: Odd functions

6.1 Scalar/Dot Product ♥

1. Define the scalar product for two vectors, \(\mathbf{u}\) and \(\mathbf{v}\), where: \[\mathbf{u} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}\]

2. Define the scalar product for two vectors, \(\vec{u}\) and \(\vec{v}\), where: \[\vec{u} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}, \quad \vec{v} = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}\]

3. Define the magnitude of a position vector \(\mathbf{u}\), where: \[\mathbf{u} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\]

4. Define the angle between two vectors \(\mathbf{u}\) and \(\mathbf{v}\), in terms of the scalar product, and the magnitude of each vector, \(|\mathbf{u}|\) and \(|\mathbf{v}|\).

6.2 Vector Line Equations ♣

With respect to a fixed origin, the lines \(l_1\) and \(l_2\) are given by the equations:

\[\mathbf{l_1} = \begin{pmatrix} -9 \\ 0 \\ 10 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\]

\[\mathbf{l_2} = \begin{pmatrix} 3 \\ 5 \\ 17 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ 5 \end{pmatrix}\]

1. Show that \(l_1\) and \(l_2\) meet, and find the position vector of their intersection.
2. Show that \(l_1\) and \(l_2\) are perpendicular.

6.3 Vector/Cross Product ♠

1. Define the vector cross product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\).

2. Define the magnitude of the vector cross product \(|\mathbf{u} \times \mathbf{v}|\) in terms of the angle between the two vectors and the magnitude of the vectors.

3. Calculate the vector perpendicular to both: \[\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 6 \\ 7 \end{pmatrix}\] and the angle between the two vectors. Ensure it matches the value obtained from the dot product.

4. Using the right hand rule for cross products, determine the direction of the cross product of a vector in the \(x\)-direction and a vector in the \(y\)-direction.

6.4 Electromagnetic Waves - A more involved example ♠

1. The electric field component is given by \(\mathbf{E} = \mathbf{E_{max}} \cos(kx-wt+\phi)\) where the electric field amplitude \(\mathbf{E_{max}}\) is given by: \[\mathbf{E_{max}} = \begin{pmatrix} 0 \\ E_y \\ 0 \end{pmatrix}\] Sketch this on a 3D set of axes.

2. The magnetic field component is given by \(\mathbf{B} = \mathbf{B_{max}} \cos(kx-wt+\phi)\) where the magnetic field amplitude \(\mathbf{B_{max}}\) is given by: \[\mathbf{B_{max}} = \begin{pmatrix} B_x \\ 0 \\ B_z \end{pmatrix}\] Sketch this on the same set of axes, taking particular care over the angle between the two waves. How could you find the angle between the two waves?

3. Using the cross product, determine the direction and magnitude of \(\mathbf{E} \times \mathbf{B}\). How does this relate to the wave-vector \(\vec{k}\) (the direction of the wave)?

4. The Poynting Vector is given by \(\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B}\), and describes the instantaneous power radiated from an electromagnetic wave. In what direction does this vector travel? What is the value of \(\cos(kx-wt+\phi)\) when the maximum value is attained?

Figure: Electromagnetic Plane Wave Diagram

7.1 Matrix Dimensions ♠

State the dimensions of each of the following matrices:

1. \(\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\)

2. \(\mathbf{B} = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\)

3. \(\mathbf{C} = \begin{pmatrix} a & b & c \\ d & e & f \end{pmatrix}\)

4. \(\mathbf{D} = \begin{pmatrix} a & b \\ c & d \\ e & f \end{pmatrix}\)

5. \(\mathbf{E} = \begin{pmatrix} a \end{pmatrix}\)

7.2 Matrix Multiplication ♠

1. Given two matrices, of dimensions \(a \times b\) and \(m \times n\) respectively, state the conditions necessary for matrix multiplication to be possible. What is the dimensions of the resulting product matrix?

2. Is matrix multiplication possible for the following matrices, of dimension:

   1. \(2 \times 3\) and \(3 \times 2\)?
   2. \(3 \times 3\) and \(3 \times 2\)?
   3. \(3 \times 2\) and \(3 \times 2\)?
   4. \(3 \times 2\) and \(2 \times 3\)?

   If it is possible, state the dimensions of the resulting matrix.

7.3 Matrix Arithmetic ♠

Find both the sum and product of the following matrices: Part (d) is character building: how good is your handwriting?

1. \(\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}\)

2. \(\mathbf{C} = \begin{pmatrix} 5 & 3 & 5 \\ 6 & 5 & 2 \\ 7 & 5 & 6 \end{pmatrix}\) and \(\mathbf{D} = \begin{pmatrix} 1 & 3 & 9 \\ -6 & -1 & -8 \\ 3 & 2 & 1 \end{pmatrix}\)

3. \(\mathbf{E} = \begin{pmatrix} 3 & 4 & 5 \\ 4 & 5 & 3 \\ 5 & 3 & 4 \end{pmatrix}\) and \(\mathbf{F} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)

4. \(\mathbf{G} = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\) and \(\mathbf{H} = \begin{pmatrix} \alpha & \beta & \gamma \\ \delta & \epsilon & \zeta \\ \eta & \theta & \kappa \end{pmatrix}\)

7.4 Determinants ♠

Find the determinants of the following matrices:

1. \(\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\)

2. \(\mathbf{B} = \begin{pmatrix} 2 & 4 \\ 4 & 2 \end{pmatrix}\)

3. \(\mathbf{C} = \begin{pmatrix} 8 & 4 \\ 4 & 2 \end{pmatrix}\)

4. \(\mathbf{D} = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\)

5. \(\mathbf{E} = \begin{pmatrix} 6 & 1 & 1 \\ 4 & -2 & 5 \\ 2 & 8 & 7 \end{pmatrix}\)

8.1 Complex Quadratics ♠

Solve the following quadratic equations:

1. \(x^2 + 4 = 0\)
2. \(2x^2 + 6x - 2 = 0\)
3. \(-3x^2 + 6x - 11 = 0\)

8.2 ‘Realising’ the denominator ♠

Much like surds, we can remove any complex numbers from the denominator, by multiplying both ‘top and bottom’ by the conjugate.

1. Express the following fractions in the form \(z = x + iy\):
   1. \(\frac{7 + 2i}{2 + 3i}\)
   2. \(\frac{a + bi}{c + di}\)
   3. \(\frac{7 + 4i}{3 - 2i}\)
2. Solve the equation for \(z\): \[(3-2i)(z-1+4i) = 7+4i\]

8.3 Argand Diagrams ♠

1. From the definition of \(z = x+iy\) and using Euler’s Identity and the polar form, prove that a complex number can be expressed in ‘modulus-argument’ form, i.e \(z = re^{i\theta}\). Give expressions for \(r = |z|\) and \(\theta\).

2. Plot the following points on the Argand Diagram. Calculate the argument and modulus of each point, and express in the form \(z = re^{i\theta}\):

   1. \(z = 1\)
   2. \(z = i\)
   3. \(z = \pi i\)
   4. \(z = 2 - 2i\)
   5. \(z = 5\sqrt{3} - 5i\)

8.4 Trigonometric Functions in terms of complex exponentials ♠

1. Given that \(z = x + iy\), show that \(z = e^{i\theta} = \cos(\theta) + i\sin(\theta)\).

2. From the modulus-argument form above, for \(e^{i\theta}\), prove that \(e^{-i\theta} = \cos(\theta) - i\sin(\theta)\).

3. Hence, by solving the equations simultaneously, prove that both: \[\cos(\theta) = \frac{1}{2}(e^{i\theta} + e^{-i\theta})\] \[\sin(\theta) = \frac{1}{2i}(e^{i\theta} - e^{-i\theta})\]

A2.1 Graph sketching ♥

e.g Wolfram Alpha, GeoGebra or Desmos Graphing Calculator can provide a check. I recommend Desmos or GeoGebra for any graphing sketching.

A3.1 Factorisation ♥

1. \((x-2)(x-1)\)
2. \(3x(x-2)(x+1)\)
3. \([x-\frac{1}{2}(1-\sqrt{5})][x-\frac{1}{2}(1+\sqrt{5})]\)
4. \((x-1)(x^2 + x + 1)\)
5. \((x-3)(x-1)^2(x+2)\)

A3.2 More Factorisation ♥

\(x < -1\) or \(0 < x < 3\)

A3.3 Partial Fractions ♥

1. \(\frac{1}{x-1} - \frac{1}{x+1}\)
2. \(\frac{1}{3}(\frac{1}{x+1} - \frac{x-2}{x^2-x+1})\)
3. \(\frac{1}{(x+1)^3} - \frac{1}{x-1} + \frac{1}{x-2}\)
4. \(1 - \frac{1}{x-2} + \frac{2}{x+1}\)

A3.4 Completing the Square ♥

Smallest when \(\sin(x) = -1\) a) 5 b) 2 c) -3

A3.5 Binomial Expansion ♣

1. 1. \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}\)
   2. \((1+x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k\)
   3. \((1+X)^n = \sum_{k=0}^{n} \binom{n}{k} X^k\)
2. \(\frac{3^k 2^{10-k} 10!}{k!(10-k)!}\)
3. \(1 + 6x + 21x^2 + 50x^3 + 90x^4\)
4. \(1 - 6x + 15x^2 - 26x^3 + 51x^4\)
5. \(\sqrt{2}(1 + \frac{1}{4}x - \frac{1}{32}x^2 + \frac{1}{128}x^3)\)

A3.6 Taylor Series ♠

2. The general form given in A3.5(aiii) is a Taylor series. When \(n=-1\), we obtain an alternating sum looking similarly \((1+x)^n\).

A4.1 Small Angle Approximations ♠

1. \(\sin(x) \approx x\)
2. \(\cos(x) \approx 1 - \frac{x^2}{2}\)
3. \(\tan(x) \approx x\)

A4.2 Trigonometric Equations ♥

1. ♥ \(n\pi + (-1)^n \frac{\pi}{6}\)
2. ♠ \(n\frac{\pi}{2} + \frac{\pi}{4}\) and \(2n\pi \pm \frac{2\pi}{3}\)

A4.3 Simple Pendulum ♣

1. We obtain that \(h = L - L\cos(\theta)\). By factorising out \(L\) and using the small angle approximation for \(\cos(\theta)\), we find the result.
2. General form of SHM: \(\ddot{x} = kx\). Using small angle approximation for \(\sin(\theta)\), we obtain the differential equation of SHM.

A5.1 Direct Differentiation ♥

1. \(20(5x+2)^3\)
2. \(3\cos(3x+2)\)
3. \(4e^{4x-1}\)
4. \(-10\sin(2x+3)\)
5. \(-3\sin(x)\cos^2(x)\)
6. \(\frac{4}{4x-5}\)

A5.2 Chain and Product Rule ♣

1. \(2x\alpha\cos(x^2)\)
2. \(\beta \frac{x^{\beta-1} - x^{-\beta-1}}{x^\beta + x^{-\beta}}\)

A5.3 Stationary Points and Graph Sketching ♥

Maximum at \((a, \frac{1}{2a})\), minimum at \((-a, -\frac{1}{2a})\)

A6.1 Indefinite Integrals

♥ a) \(\frac{x^5}{5} + c\) b) \(-3\cos(x) + c\) c) \(4e^x + c\) d) \(\frac{1}{\ln(a)} a^x + c\) e) \(-\frac{x^{-3}}{3} + c\) f) \(2\tan(x) + c\)

♣ a) \(\frac{1}{\sqrt{5}} \arctan(\frac{x-1}{\sqrt{5}}) + c\) b) \(\frac{1}{6} \ln(\frac{x^2+x+1}{(x-1)^2}) + \frac{1}{\sqrt{3}} \arctan(\frac{2x+1}{\sqrt{3}}) + c\) c) \(x\arctan(x) + \frac{1}{2} \ln(1+x^2) + c\) d) \(\frac{1}{2}(x^2-1)e^{x^2} + c\)

A6.2 Odd/Even functions ♠

All integrals are 0 since all are odd functions.

A7.1 Scalar/dot products ♥

1. \(\mathbf{u} \cdot \mathbf{v} = x_1y_1 + x_2y_2 = \sum_{i=1}^{2} x_i y_i\)
2. \(\vec{u} \cdot \vec{v} = x_1y_1 + x_2y_2 + \ldots + x_ny_n = \sum_{i=1}^{n} x_i y_i\)
3. \(|\mathbf{u}| = \sqrt{x^2 + y^2 + z^2}\)
4. \(\theta = \cos^{-1}\frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}\)

A7.2 Vector Line equation ♣

1. Meet when \(\lambda = 3\) and \(\mu = -2\) and the intersection is at \(\mathbf{l_2} = \begin{pmatrix} -3 \\ 3 \\ 7 \end{pmatrix}\)
2. Show the dot product of \(\mathbf{l_1} \cdot \mathbf{l_2} = 0\). Thus \(0 = |\mathbf{l_1}||\mathbf{l_2}|\cos(\theta) = 0\) and hence \(\cos(\theta) = 0\) \(\Rightarrow \theta = 90° \Rightarrow\) perpendicular.

A7.3 Vector/cross products ♠

1. \(\mathbf{u} \times \mathbf{v} = \begin{pmatrix} u_yv_z - u_zv_y \\ u_zv_x - u_xv_z \\ u_xv_y - u_yv_x \end{pmatrix}\)
2. \(|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin(\theta)\)
3. \(\mathbf{a} \times \mathbf{b} = \begin{pmatrix} -3 \\ 6 \\ -3 \end{pmatrix}\) Angle between vectors is \(7.47°\), thankfully matching our dot product!
4. Taking standard Cartesian axes, we have that the cross product of a vector in the \(x\) direction and a vector in the \(y\) direction will be in a direction perpendicular to both the \(x\) and \(y\) axes: i.e will be facing in the positive \(z\) direction, by the right hand rule.

A7.4 Electromagnetic Waves ♠

1. A sinusoidal graph, oriented along the y-axis.
2. A sinusoidal graph, oriented along the z-axis. Note how these are perpendicular to each other. See any EM plane wave diagram for clarification. Be wary of the choice of direction of the field amplitude.
3. The direction of this \(\mathbf{E} \times \mathbf{B}\) is along the \(x\) axis. This matches the direction of the propagating wave, and hence matches the wave-vector direction.
4. \(\mathbf{S}\) will travel in the same direction as \(\mathbf{E} \times \mathbf{B}\) and hence in the same direction as found before. The maximum value is obtained when \(\cos(\theta) = 1\).

A8.1 Matrix Dimensions ♠

1. \(2 \times 2\)
2. \(3 \times 3\)
3. \(2 \times 3\)
4. \(3 \times 2\)
5. \(1 \times 1\)

A8.2 Matrix Multiplication ♠

1. For matrix multiplication, we need that the ‘column rank’ of the first matrix is equal to the ‘row rank’ of the second matrix, i.e \(b = m\) in this case. The resulting matrix has dimensions of \(a \times n\).
2. 1. Yes: \(2 \times 2\)
   2. Yes: \(3 \times 2\)
   3. No
   4. Yes: \(3 \times 3\)

A8.3 Matrix Arithmetic ♠

1. \(\mathbf{A+B} = \begin{pmatrix} a+\alpha & b+\beta \\ c+\gamma & d+\delta \end{pmatrix}\) and \(\mathbf{AB} = \begin{pmatrix} a\alpha+b\gamma & a\beta+b\delta \\ c\alpha+d\gamma & c\beta+d\delta \end{pmatrix}\)

2. \(\mathbf{C+D} = \begin{pmatrix} 6 & 6 & 14 \\ 0 & 4 & -6 \\ 10 & 7 & 7 \end{pmatrix}\) and \(\mathbf{CD} = \begin{pmatrix} 2 & 22 & 26 \\ -18 & 7 & 16 \\ -5 & 28 & 29 \end{pmatrix}\)

3. \(\mathbf{E+F} = \begin{pmatrix} 4 & 4 & 5 \\ 4 & 6 & 3 \\ 5 & 3 & 5 \end{pmatrix}\) and \(\mathbf{EF} = \begin{pmatrix} 3 & 4 & 5 \\ 4 & 5 & 3 \\ 5 & 3 & 4 \end{pmatrix}\)

4. \(\mathbf{G+H} = \begin{pmatrix} a+\alpha & b+\beta & c+\gamma \\ d+\delta & e+\epsilon & f+\zeta \\ g+\eta & h+\theta & i+\kappa \end{pmatrix}\) and \(\mathbf{GH} = \begin{pmatrix} a\alpha+b\delta+c\eta & a\beta+b\epsilon+c\theta & a\gamma+b\zeta+c\kappa \\ d\alpha+e\delta+f\eta & d\beta+e\epsilon+f\theta & d\gamma+e\zeta+f\kappa \\ g\alpha+h\delta+i\eta & g\beta+h\epsilon+i\theta & g\gamma+h\zeta+i\kappa \end{pmatrix}\)

A8.4 Determinants ♠

1. \(|\mathbf{A}| = ad - bc\)
2. \(|\mathbf{B}| = -12\)
3. \(|\mathbf{C}| = 0\)
4. \(|\mathbf{D}| = a\begin{vmatrix} e & f \\ h & i \end{vmatrix} - b\begin{vmatrix} d & f \\ g & i \end{vmatrix} + c\begin{vmatrix} d & e \\ g & h \end{vmatrix}\) Explicitly, \(|\mathbf{D}| = a(ei - fh) - b(di - fg) + c(dh - eg)\)
5. \(|\mathbf{E}| = -306\)

A9.1 Complex Quadratics ♠

1. \(x = \pm 2i\)
2. \(x = -\frac{3}{2} \pm \frac{\sqrt{13}}{2}\)
3. \(x = 1 \pm i\frac{2\sqrt{6}}{3} = 1 \pm 2i\sqrt{\frac{2}{3}}\)

A9.2 ‘Realising’ the denominator ♠

1. Realising the fractions:
   1. \(\frac{20 - 17i}{13} = \frac{20}{13} - i\frac{17}{13}\)
   2. \(\frac{ac+bd+i(bc-ad)}{c^2+d^2} = \frac{ac+bd}{c^2+d^2} + i\frac{bc-ad}{c^2+d^2}\)
   3. \(1 + 2i\)
2. Solving the complex equation: \(2(1+i)\)

A9.3 Argand Diagrams ♠

1. Essentially given in the preamble at the start of the Complex section.
2. 1. \(\theta = 0\), \(r = 1\) \(\Rightarrow\) \(z = 1\)
   2. \(\theta = \frac{\pi}{2}\), \(r = 1\) \(\Rightarrow\) \(z = e^{i\frac{\pi}{2}}\)
   3. \(\theta = \frac{\pi}{2}\), \(r = \pi\) \(\Rightarrow\) \(z = \pi e^{i\frac{\pi}{2}}\)
   4. \(\theta = \frac{3\pi}{4}\), \(r = \sqrt{8}\) \(\Rightarrow\) \(z = \sqrt{8} e^{i\frac{3\pi}{4}}\)
   5. \(\theta = \frac{5\pi}{3}\), \(r = 10\) \(\Rightarrow\) \(z = 10 e^{i\frac{5\pi}{3}}\)

A9.4 Trigonometric Functions ♠

1. Substitute the polar form in, see 3)a).
2. Use the odd/even properties of \(\sin(\theta)\) and \(\cos(\theta)\).
3. Eliminate each variable separately.

Graphing

The equation of a straight-line graph is given as \(y - y_1 = m(x - x_1)\) where \(m\) is the gradient and \(x, x_1, y\) and \(y_1\) are points on the straight line.

Two straight lines are perpendicular if the product of their gradients \(m_1\) and \(m_2\) is \(-1\) (i.e \(m_1m_2 = -1\)).

Quadratic Equations

\(ax^2 + bx + c = 0\) has the root \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\)

Law of Indices

• \(a^x \cdot a^y = a^{x+y}\)
• \(a^x \div a^y = a^{x-y}\)
• \((a^x)^y = a^{xy}\)

Trigonometry

• \(\cos^2(x) + \sin^2(x) \equiv 1\)
• \(\sec^2(x) \equiv 1 + \tan^2(x)\)
• \(\csc^2(x) \equiv 1 + \cot^2(x)\)
• \(\sin(2x) \equiv 2\sin(x)\cos(x)\)
• \(\cos(2x) \equiv \cos^2(x) - \sin^2(x)\)

Differentiation

Integration

Vectors

\(|x\mathbf{i} + y\mathbf{j} + z\mathbf{k}| = \sqrt{x^2 + y^2 + z^2}\)

Consider two vectors \(\mathbf{u} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}\)

The scalar product of these vectors is: \[\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \cdot \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix} = x_1y_1 + x_2y_2 + x_3y_3 = |\mathbf{u}||\mathbf{v}|\cos\theta\]

where \(\theta\) is the acute angle between the vectors \(\mathbf{u}\) and \(\mathbf{v}\).

The equation of the line through the point with position vector \(\mathbf{u}\) parallel to the vector \(\mathbf{v}\) is given as \(\mathbf{r} = \mathbf{u} + t\mathbf{v}\).

The equation of the plane containing the point with position vector \(\mathbf{u}\) and perpendicular to vector \(\mathbf{w}\) is \((\mathbf{r} - \mathbf{a}) \cdot \mathbf{w} = 0\).

Matrices

Given a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\).

The determinant, \(\Delta = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc\).

The inverse is \(\frac{1}{\Delta} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\).

For matrices \(\mathbf{A}, \mathbf{B}\): \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\)

Complex Numbers

\(i = \sqrt{-1}\)

Consider two complex numbers \(z_1 = r_1e^{i\theta_1}\) and \(z_2 = r_2e^{i\theta_2}\),

\(z_1z_2 = r_1r_2e^{i(\theta_1+\theta_2)}\).

\(\frac{z_1}{z_2} = \frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}\)

Exponential: \(e^{i\theta} = \cos\theta + i\sin\theta\) \(e^{-i\theta} = \cos\theta - i\sin\theta\)

Adapted July 2018
Brad Din
Updated August 2023
Dr. Oto-obong Inyang