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Skill 12 of 13
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Vectors

Mathematical skill ●●●○○

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

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

Whenever you hear someone say a position as a single number without also saying the direction – ‘Where did they go? 10 more metres and you’ll find them!’ – then the direction must be implied. This single number itself is just the “size” of the position, the ‘how far’ part. We could call it the magnitude of the vector. It can be thought of as the length of the position vector arrow.2

Let us invent an arrow symbol ~^\to to symbolise a vector. We can simply add it to the s symbol we already chose for position:3

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

The s is a position where the direction is implied, and the \overset{\to}{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 or length is its magnitude.

To aid our writing let us agree that coordinates can also be written horizontally. We will just have to separate them by something, for example a comma , :4

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

Whether it is written vertically or horizontally (and comma-separated) it is still a vector.

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.[5,6,7]

So:

  • length or distance L, area A, volume V, time t, quantity n and of course the directionless position s are scalars, carrying magnitude only, while
  • \overset{\to}{s} is a vector, carrying magnitude and direction.5

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


References:

  1. Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com
  2. What Is a Vector?’ (web page), Steven Holzner, For Dummies, A Wiley Brand, www.dummies.com/education/science/physics/what-is-a-vector (accessed Nov. 2nd, 2020)
  3. An introduction to vectors’ (web page), David Frank & Duane Q. Nykamp, Math Insight, www.mathinsight.org/vector_introduction (accessed Nov. 2nd, 2020)
  4. 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
  5. Scalars and Vectors’ (web page), The Physic Classroom, www.physicsclassroom.com/class/1DKin/Lesson-1/Scalars-and-Vectors (accessed Sep. 12th, 2019)
  6. Scalar’ (encyclopedia), Encyclopædia Britannica, 2009, www.britannica.com/science/scalar (accessed May 7th, 2019)
  7. Sears and Zemansky’s Univesity Physics with Modern Physics’ (book), Hugh D. Young & Roger A. Freedman, Pearson Education, 13th ed., 2012