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Last Friday David Bowman asked:
... What do you tell an inquisitive student *what it means*
that a vector points in the x-direction with magnitude 1?
Or even (forget about the magnitude for now) just what
*it means* that a vector points in a given direction? What
does pointing in a direction *mean*? ...
In my opinion the answer is not more difficult than assigning
meaning to "one real number being larger than another". What
does the phrase "3 Lb of lead is larger than 2 Lb of wood"
mean in science? I would say it means the scale reading number
for lead is larger than for wood. But in mathematics, where we
deal with pure numbers I would say, for example, that 3 is
larger than 2 "by definition".
One-dimensional space (call it x axis) is defined by an arbitrary
line in our old 3 dimensional space. Real numbers are associated
with that axis. They increase as we move in one direction and
they decrease as we move in the opposite direction. An arrow
head is used to indicated this.
Vectors are segments of the axis with heads and tails (like arrows).
A unit vector is defined, on the axis, by a pair of numbers, they
give meaning to its direction. A vector "points along the axis"
when its head is at a higher number than its tail. Why is this trivial
answer not 100% sufficient in Euclidian space?