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Skill 12 of 13
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Vectors

Let’s give a name to properties that are collections of numbers: let’s call them vectors.1

Position is hereby a vector. We can call it the position vector, when we wish to emphasise that fact. Vectors somewhat feel like arrows, and we can draw a position as an arrow to the location.

Whenever you hear someone tell a position with only a number without also telling the direction – ‘where did they go? 10 more metres and you’ll find them!’ – the direction must be implied. This number itself is clearly just the “size” of the position, the ‘how far’ part. We could call it the magnitude of the vector, since it sort of feels like the length of that arrow we just drew.

Let’s invent an arrow symbol $\vec ~$ to symbolise a vector. We can simply add it to the $s$ symbol we already chose for position:2

$s=5\,\mathrm{m}\quad$ and $\quad \vec s= \left(\begin{matrix}
3 \\
4 \\
\end{matrix}\right)\mathrm {m}$

The $s$ is a position, where the direction is implied. And the $\vec s$ is also a position. Both are positions. Obviously, position without its direction is exactly the same as distance or length $L$. This size property fully overlaps with $s$. So, you will often hear people refer to the magnitude of position as the distance or length of position. You might even hear people say: ‘position is a vector; distance/length is its magnitude’.

To ease our writing, let’s agree that coordinates can also be written horizontally if we need to – we’ll just have to separate them by something, for example a comma $,$ :3

$(3,4)\,\mathrm{m}\quad$ and $\quad(3,4,5)\,\mathrm{m}$

We have now invented the fine word ‘vector’ for a property that is a collection of numbers. Similarly, a fine word for a property that is a single number also exists: it is called a scalar.[3,4,5]

So,

  • length/distance $L$, area $A$, volume $V$, time $t$, quantity $n$ and of course the direction-less position $s$ are scalars, carrying magnitude only, while
  • $\vec s$ is a vector, carrying magnitude and direction.4

We will rarely use the word ‘scalar’, but you can easily encounter it in technical surroundings.


References:

  1. Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com
  2. Quantities and units -- Part 2: Mathematical signs and symbols to be used in the natural sciences and technology’ (article), ISO 80000-2:2009, 1st ed., 2009, www.iso.org/standard/31887.html
  3. Scalars and Vectors’ (web page), The Physic Classroom, www.physicsclassroom.com/class/1DKin/Lesson-1/Scalars-and-Vectors (accessed Sep. 12th, 2019)
  4. Scalar’ (encyclopedia), Encyclopædia Britannica, 2009, www.britannica.com/science/scalar (accessed May 7th, 2019)
  5. Sears and Zemansky’s Univesity Physics with Modern Physics’ (book), Hugh D. Young & Roger A. Freedman, Pearson Education, 13th ed., 2012