The Mathematical Toolkit

Physics borrows its grammar from mathematics. The first word in that grammar is the vector: a quantity that only makes sense once you know which way it points.

Throughout this course we will lean on algebra, trigonometry and geometry, plus three tools that may be newer to you: vector algebra, differential calculus and integral calculus. This chapter builds all three from the ground up, and finishes with how to report a measurement honestly.

Numbers that just add#

Pour 250 g of milk into a mug that already holds 150 g of coffee. The mug now contains 400 g of liquid. No geometry is involved; the masses add like ordinary numbers. A quantity that is fully described by a single value (with its unit) and that combines by ordinary arithmetic is called a scalar.

Mass, time, temperature, energy and volume are all scalars.

Numbers that need a direction#

Now think about velocity. Saying a cyclist moves at 5 m/s tells you how quickly, but not where. North? Up a hill? Toward you? To describe velocity completely you need a size and a direction.

The natural picture is an arrow. Choose a scale (say 1 cm of paper for every 1 m/s), draw a line of the right length parallel to the motion, and put an arrowhead on the leading end. The pointed end is the head, the other end the tail.

Here is the surprising part. Direction changes how such quantities combine. Try it below: two walks of fixed length can leave you anywhere between 1 m and 7 m from where you started.

Two motions at once#

Imagine a bead that slides along the inside of a drinking straw at 6 cm/s. While it slides, you push the whole straw sideways across the table at 8 cm/s, keeping it perpendicular to its length. After one second the bead has gone 6 cm along the straw and the straw has shifted 8 cm sideways. Watch where the bead actually ends up relative to the table.

The bead travels in a straight diagonal line: 10 cm in one second, at about 53° from the straw. Its true velocity is 10 cm/s, not 6 + 8 = 14 cm/s. Those three numbers (6, 8, 10) form a right triangle, and that is no coincidence.

The resultant is also called the sum of the two arrows, and this way of adding is the triangle rule of addition.

In print we mark vectors with an arrow on top, as in $\vec{AB}$, $\vec{v}$ or $\vec{F}$, or with bold type: v, F. The plain letter $v$ (or $|\vec v|$) then means just the magnitude.

When are two vectors equal?#

Two vectors representing the same kind of quantity are equal when they have the same magnitude and the same direction. Where you draw them does not matter. Slide an arrow anywhere on the page without rotating or stretching it, and it is still the same vector. This freedom to translate arrows is exactly what lets us move the second arrow to the head of the first when we add.

Adding vectors is a drawing exercise before it is an algebra exercise. Get the picture right and the formula falls out of a single right triangle.

Two ways to draw the same sum#

To add $\vec a$ and $\vec b$ with the triangle rule, place the tail of $\vec b$ on the head of $\vec a$. The arrow from the tail of $\vec a$ to the head of $\vec b$ is the sum.

The parallelogram rule says the same thing differently. Draw both vectors from a common tail, complete the parallelogram they span, and the diagonal from the shared tail is the sum. It is the triangle rule seen twice: the far sides of the parallelogram are just copies of $\vec a$ and $\vec b$, slid into place.

How long is the sum?#

Let the magnitudes be $a$ and $b$, with angle $\theta$ between the vectors when drawn tail to tail. In the head-to-tail picture, extend the line of $\vec a$ and drop a perpendicular from the far corner D to meet it at E.

Side BE is the part of $\vec b$ that runs along $\vec a$: $b\cos\theta$. Side DE is the part that sticks out sideways: $b\sin\theta$. Pythagoras on the right triangle ADE gives

and the last bracket is exactly 1. So:

The same triangle also tells us the direction. If $\alpha$ is the angle the sum makes with $\vec a$, then opposite over adjacent in triangle ADE gives

Check the extreme cases in the graph below. At $\theta = 0$ the cosine is 1 and the sum is simply $a + b$. At $\theta = 180^\circ$ the cosine is −1 and you get $|a - b|$. At $90^\circ$ the middle term disappears and you are back to Pythagoras.

Multiply an arrow by a number and it stretches. Multiply by a negative number and it turns around.

Scaling a vector#

Take a vector $\vec a$ of magnitude $a$ and an ordinary number $k$. The product $\vec b = k\vec a$ is defined to be a vector of magnitude $|k|\,a$:

• if $k > 0$, it points the same way as $\vec a$;
• if $k < 0$, it points the opposite way.

Multiplying by −1 keeps the length and reverses the direction. So $\vec a$ and $-\vec a$ are equally long and point in opposite directions.

This split into “how much” and “which way” comes up constantly. In the next lesson the three unit vectors $\hat\imath, \hat\jmath, \hat k$ along the coordinate axes will turn vector problems into arithmetic.

Subtracting a vector#

Subtraction needs no new rule. We define $\vec a - \vec b$ as the sum of $\vec a$ and $-\vec b$:

Because $-\vec b$ makes an angle of $180^\circ - \theta$ with $\vec a$, the magnitude formula changes only by a sign:

Shine a light straight down on an arrow and look at its shadow. That shadow is a component, and components turn vector geometry into plain arithmetic.

Resolving a vector in a plane#

Put a vector $\vec a = \vec{OA}$ in the x–y plane with its tail at the origin. Say it makes angle $\alpha$ with the x-axis and $\beta$ with the y-axis. Drop perpendiculars from A to each axis, meeting them at B and C. By the triangle rule,

We have split one arrow into two: one lying along x, one along y. The length OB is the projection of $\vec a$ on the x-axis, which is $a\cos\alpha$. Likewise OC $= a\cos\beta$.

Now bring in the unit vectors $\hat\imath$ and $\hat\jmath$, arrows of length 1 pointing along +x and +y. Then $\vec{OB} = a\cos\alpha\,\hat\imath$ and $\vec{OC} = a\cos\beta\,\hat\jmath$, so

Three dimensions#

A vector that leaves the x–y plane also casts a shadow on the z-axis. If it makes angles $\alpha, \beta, \gamma$ with the x, y and z axes, then with $\hat k$ the unit vector along z:

The numbers $a_x = a\cos\alpha$, $a_y = a\cos\beta$, $a_z = a\cos\gamma$ are the components of $\vec a$. Equation (2.3) says that every vector is a combination of the same three building blocks $\hat\imath, \hat\jmath, \hat k$.

More generally, the component of $\vec a$ along any direction that makes angle $\theta$ with it is $a\cos\theta$, which is its projection onto that direction. Nothing forces us to use x, y, z. Physicists often choose axes along a slope, along a rope, or along the motion, whichever makes the problem easiest.

Why components make addition easy#

Suppose we know three vectors by their components:

Adding them is just collecting like terms:

No triangles, no cosine rule, however many vectors you add. Try it with three arrows:

For vectors in the x–y plane the result is two perpendicular pieces, $R_x = a_x+b_x+c_x$ and $R_y = a_y+b_y+c_y$, so

Two arrows go in, a single number comes out. The dot product measures how much two vectors agree about direction.

Definition#

The dot product (or scalar product) of $\vec a$ and $\vec b$ is

Read it as “the length of $\vec a$ times the shadow of $\vec b$ on $\vec a$” (or the other way round, since the formula is symmetric). Explore it:

Three cases are worth fixing in memory:

• Same direction (θ = 0): $\vec a\cdot\vec b = ab$, the largest possible value.
• Perpendicular (θ = 90°): $\cos 90^\circ = 0$, so the dot product is zero. This is the standard test for perpendicularity.
• Opposite (θ = 180°): $\vec a\cdot\vec b = -ab$.

Algebraic rules#

The dot product behaves like ordinary multiplication in two important ways:

The dot product from components#

Write $\vec a = a_x\hat\imath + a_y\hat\jmath + a_z\hat k$ and similarly for $\vec b$. Expanding the product with the distributive law gives nine terms, each a dot product of two unit vectors. Those are easy, because the unit vectors have length 1 and are mutually perpendicular:

Six of the nine terms vanish, leaving a formula you will use constantly:

The second way to multiply vectors produces a new vector, one that stands perpendicular to both of the originals. Torque, angular momentum and magnetic force are all built this way.

Magnitude#

The cross product (or vector product) $\vec a\times\vec b$ is a vector whose size is

Draw the two vectors tail to tail and you see two angles between them, one less than 180° and one more (they add to 360°). The θ here is always the smaller one, so $\sin\theta \ge 0$ and the magnitude is never negative.

Geometrically, $ab\sin\theta$ is the area of the parallelogram spanned by the two vectors (base $a$ times height $b\sin\theta$). Parallel or anti-parallel vectors span no area, so their cross product is zero.

Direction: the right-hand rule#

$\vec a\times\vec b$ points perpendicular to the plane containing $\vec a$ and $\vec b$. That leaves two choices, up or down, and a convention picks one:

Push $\varphi$ past 180° in the lab above. The smaller angle is now measured the other way round, your fingers curl the opposite way, and the product flips downward. Hit “swap order” and the same thing happens. Reversing the order of a cross product reverses its direction:

It does distribute over addition, but it is not associative:

Right-handed axes#

Once you choose x and y axes, the z-axis must be perpendicular to both, but it could point either way. We always choose it so that $\hat\imath\times\hat\jmath = \hat k$. Such a frame is called right-handed, and everything else follows from the cycle:

Cross product from components#

Expand $(a_x\hat\imath + a_y\hat\jmath + a_z\hat k)\times(b_x\hat\imath + b_y\hat\jmath + b_z\hat k)$ into nine terms and use the cycle above. Three terms vanish (a unit vector crossed with itself) and the rest pair up:

Physics is full of “how fast” questions: how fast position changes, how fast velocity changes, how fast temperature drops. The derivative is the precise answer.

Rate of change, roughly#

Take two quantities $y$ and $x$ linked so that each $x$ gives exactly one $y$, and plot the graph. Nudge $x$ by a small amount $\Delta x$; $y$ responds with a change $\Delta y$. A natural first guess for “the rate at which $y$ changes with $x$” is

which is the slope of the straight chord (the secant) joining A $(x, y)$ to B $(x + \Delta x,\ y + \Delta y)$.

The trouble is that a curve’s steepness changes along the way. Between A and B it might start gentle and end steep, and the chord averages all of that into one number. To pin down the rate at A itself we need B closer. And closer.

As $\Delta x$ shrinks the chord pivots about A and settles onto the tangent line, and the ratio $\Delta y/\Delta x$ settles on a definite value even though both $\Delta y$ and $\Delta x$ are heading to zero. That limiting value is the derivative.

For a small but finite change you can turn this around and estimate: $\Delta y \approx \dfrac{dy}{dx}\,\Delta x$. This approximation is how physicists reason about small changes all the time.

Computing a derivative by hand#

Graphs are great for intuition, but we usually know $y$ as a formula. Take the area of a square of side $L$: $A = L^2$. Grow the side to $L + \Delta L$:

So $\dfrac{d}{dL}(L^2) = 2L$. The picture says why: the area grows mostly through two strips of length $L$, and the little corner is a second-order crumb that vanishes in the limit.

A small library of derivatives#

You will not do the limit by hand every time. Here are the results for the functions that show up most in physics. $dy/dx$ is also called the derivative or differential coefficient of $y$ with respect to $x$.

Three rules let you combine them:

The highest point of a ball’s flight, the cheapest shape for a can, the angle that throws farthest: optimisation problems all come down to finding where a slope is zero.

Flat tops and flat bottoms#

Picture any smooth curve with a hump and a dip. At the very top of the hump (a maximum) and the very bottom of the dip (a minimum), the tangent line lies perfectly flat. A flat tangent has zero slope, and slope is the derivative. So at a maximum or minimum,

Peak or valley? Ask the second derivative#

A zero slope alone does not say which one you have found. Look at how the slope behaves as you pass through the point:

• Approaching a maximum the slope is positive, at the top it is zero, and afterwards it is negative. The slope is decreasing.
• Through a minimum the slope goes from negative to zero to positive. The slope is increasing.

“How fast the slope changes” is the derivative of the derivative, written $\dfrac{d}{dx}\!\left(\dfrac{dy}{dx}\right) = \dfrac{d^2y}{dx^2}$. That gives the test:

Differentiation breaks a quantity into rates. Integration does the reverse: it adds up countless tiny contributions to rebuild the whole.

The area under a curve#

Let a curve PQ show $y = f(x)$ between $x = a$ and $x = b$. Drop vertical lines from P and Q to the x-axis, meeting it at A and B. What is the area of the region PABQ under the curve?

Here is a strategy that works for any curve. Chop AB into $N$ equal strips, each of width $\Delta x = (b - a)/N$. Over each strip, draw a rectangle whose height is the curve’s height at the strip’s left edge. Rectangles are easy to measure, so add them:

This is a little off, because thin slivers between the bar tops and the curve are missed (or overcounted). But use more, thinner strips and the bars touch the curve at more points while the slivers shrink. Try it:

As $N \to \infty$ (equivalently $\Delta x \to 0$) the error vanishes and the sum becomes the exact area. We write that limit with a stretched S, for “sum”:

Doing one by hand: the area under y = x#

Take $f(x) = x$. The region under it between $a$ and $b$ is a trapezium, so we know the answer should be (average height) × width $= \tfrac12(a+b)(b-a)$. Let’s get there by summing bars. The bar heights are $a,\ a+\Delta x,\ a + 2\Delta x, \dots$, an arithmetic progression of $N$ terms, so