Vectors and Measurement: Units, Sig Figs and Components

The SI has seven base units: the meter (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for amount of substance and candela (cd) for light intensity. Every other unit is built from these. A newton, for example, is 1 kg·m/s².

Since 2019 all seven are defined by fixed values of nature's constants. The second comes from a cesium atom, the meter from the speed of light and the kilogram from the Planck constant, so any well-equipped lab can reproduce them.

Prefixes scale a unit by powers of ten, so you can write 0.000 001 s as 1 µs or 5 000 000 W as 5 MW. The ones you'll use most are below.

Physics deals with numbers that are very large or very small. Scientific notation writes them as a number from 1 up to (but not including) 10, times a power of ten:

6 370 000 m = 6.37 × 10⁶ m    0.00052 m = 5.2 × 10⁻⁴ m

Count how many places the decimal point moves. Moving it left to make a big number small gives a positive power; moving it right to make a small number bigger gives a negative power. To multiply, multiply the front numbers and add the powers: (2 × 10³)(4 × 10⁵) = 8 × 10⁸.

No measurement is exact. When you read a ruler, you write down the digits you're sure of plus one more that you estimate. Those digits are the significant figures.

The rules for counting them:

• Nonzero digits always count. 452 has 3.
• Zeros between nonzero digits count. 100.2 has 4.
• Leading zeros never count; they only place the decimal point. 0.0034 has 2.
• Trailing zeros count if there's a decimal point. 0.00340 has 3; 2.050 × 10³ has 4.

When you multiply or divide, round the answer to the fewest significant figures among the numbers you used. When you add or subtract, round to the fewest decimal places.

Precision and accuracy are different things. A scale that always reads 2 kg too heavy gives precise readings, because they agree with each other, but they aren't accurate. Averaging many readings reduces random error, but it can't fix a scale that's set wrong.

To convert, multiply by a fraction equal to 1, with the unit you want on top and the unit you don't want on the bottom. Units cancel just like numbers:

72 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 20 m/s

Some conversions are exact by definition: 1 inch is exactly 2.54 cm, and 1 mile is exactly 1.609344 km. Always write units through every step of a calculation. If the units at the end aren't the ones you expected, you've made a mistake somewhere.

Walk 3 km east, then 4 km north. You walked 7 km, but you ended up only 5 km from where you started. The 7 km is a distance, a scalar. The 5 km in a set direction is your displacement, a vector.

To add vectors on paper, draw them tip to tail: draw the first arrow, start the second where the first ends and so on. The resultant is the arrow from the tail of the first to the tip of the last. The order doesn't matter: A + B gives the same arrow as B + A.

When the two vectors are at right angles, as in the walk, the Pythagorean theorem gives the size: √(3² + 4²) = 5 km. The direction comes from tan θ = 4/3, so θ ≈ 53° north of east.

Any vector in a plane can be split into an x-part and a y-part that add, tip to tail, back to the original. If a vector has size A and makes an angle θ with the +x axis:

Aₓ = A cos θ    Aᵧ = A sin θ

Going back, the size is A = √(Aₓ² + Aᵧ²) and the angle comes from tan θ = Aᵧ / Aₓ. Check which quadrant the vector points into, because a calculator's inverse tangent only returns angles from −90° to 90°.

Components turn vector addition into ordinary addition: add all the x-components, add all the y-components, then rebuild the resultant. This works for any number of vectors at any angles, and it's how the calculator below works.

Before the metric system, units of length and weight changed from town to town. Revolutionary France set out to build units based on nature, and over two centuries those units became the SI. The arrow notation for vectors came later, from scientists who wanted a simpler way to write physics in three dimensions.