Scalars & Vectors

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.