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Physics 1

Physical quantities, units and vectors

The SI, dimensional analysis with the pendulum example, significant figures, vector components, addition, dot and cross products with a worked example.

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Physics describes the world with numbers, but a number alone is not enough: “3” means nothing, “3 metres” means something, and “3 metres north” means even more. This first lesson sets up the tools needed for the whole course: quantities and units, dimensional analysis, vectors and their operations.

Physical quantities and the SI

A physical quantity is a property that can be measured: length, mass, time, temperature. The result of a measurement is always a number followed by a unit. The International System of Units (SI) starts from seven base quantities; all the others (velocity, force, energy, charge) are built by combining them.

QuantitySI unitSymbol
lengthmetrem
masskilogramkg
timeseconds
electric currentampereA
temperaturekelvinK
amount of substancemolemol
luminous intensitycandelacd

Since 2019 every SI unit is defined by fixing the exact value of a constant of nature: the second from a transition frequency of caesium, the metre from the speed of light (c = 299\,792\,458 m/s), the kilogram from Planck’s constant, the ampere from the electron charge.

The derived units we will use most are the newton (1\ \text{N} = 1\ \text{kg·m/s}^2), the joule (1\ \text{J} = 1\ \text{N·m}), the watt (1\ \text{W} = 1\ \text{J/s}) and the pascal (1\ \text{Pa} = 1\ \text{N/m}^2).

Dimensional analysis

Every physical equation must be homogeneous: both sides must have the same dimensions. It is the fastest check there is for catching a mistake in a calculation. We write [L], [M], [T] for the dimensions of length, mass and time.

Example. The period of a simple pendulum might depend on the length l, the mass m and g. Write T = k\, l^a m^b g^c and match the dimensions:

    \[ [T] = [L]^a [M]^b \left([L][T]^{-2}\right)^c \]

For mass b = 0, for length a + c = 0, for time -2c = 1. So c = -\tfrac12, a = \tfrac12 and T = k\sqrt{l/g}. Dimensional analysis does not give the constant k (we will see it is 2\pi), but it tells us straight away that the period does not depend on the mass and that four times the length gives twice the period.

Significant figures and uncertainty

A measurement without its uncertainty is incomplete: l = (1.253 \pm 0.002) m. In calculations the result cannot be more precise than the data: if you time something to a hundredth of a second, there is no point in writing the speed with six decimals. Rule of thumb: in products and quotients keep as many significant figures as the least precise input has.

Scalars and vectors

Mass, time, temperature and energy are scalars: a number with its unit is enough. Displacement, velocity, acceleration and force are vectors: they need a magnitude, a direction and a sense. We write \vec A (or bold) and the magnitude |\vec A| or simply A.

In a Cartesian frame each vector can be split along the axes using the unit vectors \hat\imath, \hat\jmath, \hat k, which have length 1 and point along the axes:

    \[ \vec A = A_x\,\hat\imath + A_y\,\hat\jmath + A_z\,\hat k \]

Components of a vector and the sum of two vectors with the parallelogram rule
Left: the components A_x = A\cos\theta and A_y = A\sin\theta of a vector in the plane. Right: the sum \vec A + \vec B is the diagonal of the parallelogram built on the two vectors.

In the plane, if \theta is the angle with the x axis:

    \[ A_x = A\cos\theta,\qquad A_y = A\sin\theta,\qquad A = \sqrt{A_x^2 + A_y^2},\qquad \tan\theta = \frac{A_y}{A_x} \]

Be careful with the last formula: \arctan returns an angle between -90° and +90°, so when A_x \lt 0 you have to add 180°. The atan2 function of calculators and programming languages does this for you.

Sum and difference

Geometrically, the sum is found with the parallelogram rule, or by placing the tail of the second vector on the tip of the first. With components it is even simpler: add component by component.

    \[ \vec A + \vec B = (A_x + B_x)\,\hat\imath + (A_y + B_y)\,\hat\jmath + (A_z + B_z)\,\hat k \]

The difference \vec A - \vec B is the sum of \vec A and the opposite of \vec B. Multiplying a vector by a scalar k multiplies its magnitude by |k| and, if k \lt 0, reverses its sense.

Example. A boat moves east at 4 m/s on a river whose current flows north at 3 m/s. Its velocity relative to the bank is \vec v = 4\,\hat\imath + 3\,\hat\jmath m/s, with magnitude \sqrt{16 + 9} = 5 m/s, at an angle \arctan(3/4) \approx 36.9° from east.

Dot product

The dot (scalar) product of two vectors is a number:

    \[ \vec A \cdot \vec B = A\,B\cos\varphi \]

where \varphi is the angle between them. It is largest when they are parallel, zero when they are perpendicular, negative when the angle is obtuse. In physics it appears in the work of a force, W = \vec F\cdot\vec s, and in power, P = \vec F\cdot\vec v.

Proof: the dot product in components

The unit vectors of the axes are perpendicular to each other and have length 1, so \hat\imath\cdot\hat\imath = \hat\jmath\cdot\hat\jmath = \hat k\cdot\hat k = 1 and \hat\imath\cdot\hat\jmath = \hat\jmath\cdot\hat k = \hat k\cdot\hat\imath = 0. Expanding the product with the distributive property, of the nine terms only those with equal unit vectors survive:

    \[ \vec A\cdot\vec B = A_x B_x + A_y B_y + A_z B_z \]

This formula gives the angle between two vectors known by their components: \cos\varphi = \dfrac{\vec A\cdot\vec B}{A\,B}. With \vec A = \vec B you get Pythagoras’ theorem in three dimensions, A^2 = A_x^2 + A_y^2 + A_z^2.

Cross product

The cross (vector) product \vec C = \vec A \times \vec B is a vector:

  • magnitude C = A\,B\sin\varphi, the area of the parallelogram built on the two vectors;
  • direction perpendicular to the plane of \vec A and \vec B;
  • sense given by the right-hand rule: index finger along \vec A, middle finger along \vec B, the thumb points along \vec C.

It is not commutative: \vec B\times\vec A = -\vec A\times\vec B. The product of a vector with itself is zero. In components it is computed with a determinant:

    \[ \vec A\times\vec B = (A_y B_z - A_z B_y)\,\hat\imath + (A_z B_x - A_x B_z)\,\hat\jmath + (A_x B_y - A_y B_x)\,\hat k \]

We will use it for the torque of a force, \vec\tau = \vec r\times\vec F, for angular momentum and, in Physics 2, for the Lorentz force \vec F = q\,\vec v\times\vec B, the working principle of every electric motor.

Worked example

Given \vec A = 3\,\hat\imath - 2\,\hat\jmath + \hat k and \vec B = \hat\imath + 4\,\hat\jmath - 2\,\hat k, find \vec A + \vec B, \vec A\cdot\vec B, the angle between the vectors and \vec A\times\vec B.

  • Sum: \vec A + \vec B = 4\,\hat\imath + 2\,\hat\jmath - \hat k.
  • Dot product: 3\cdot 1 + (-2)\cdot 4 + 1\cdot(-2) = 3 - 8 - 2 = -7.
  • Magnitudes: A = \sqrt{9 + 4 + 1} = \sqrt{14}, B = \sqrt{1 + 16 + 4} = \sqrt{21}; \cos\varphi = -7/\sqrt{294} \approx -0.408, so \varphi \approx 114°.
  • Cross product: \big((-2)(-2) - 1\cdot 4\big)\hat\imath + \big(1\cdot 1 - 3\cdot(-2)\big)\hat\jmath + \big(3\cdot 4 - (-2)\cdot 1\big)\hat k = 0\,\hat\imath + 7\,\hat\jmath + 14\,\hat k.

Quick check: the cross product must be perpendicular to \vec A: 3\cdot 0 + (-2)\cdot 7 + 1\cdot 14 = 0. Correct.

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