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Physicists,

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air traffic controllers,

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and video game creators

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all have at least one thing in common:

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vectors.

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What exactly are they,
and why do they matter?

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To answer,
we first need to understand scalars.

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A scalar is a quantity with magnitude.

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It tells us how much 
of something there is.

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The distance between you and a bench,

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and the volume and temperature
of the beverage in your cup

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are all described by scalars.

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Vector quantities also have a magnitude
plus an extra piece of information,

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direction.

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To navigate to your bench,

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you need to know how far away it is
and in what direction,

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not just the distance,
but the displacement.

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What makes vectors special
and useful in all sorts of fields

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is that they don't change 
based on perspective

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but remain invariant 
to the coordinate system.

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What does that mean?

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Let's say you and a friend 
are moving your tent.

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You stand on opposite sides
so you're facing in opposite directions.

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Your friend moves two steps to the right
and three steps forward

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while you move two steps to the left
and three steps back.

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But even though it seems 
like you're moving differently,

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you both end up moving 
the same distance in the same direction

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following the same vector.

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No matter which way you face,

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or what coordinate system you place
over the camp ground,

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the vector doesn't change.

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Let's use the familiar 
Cartesian coordinate system

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with its x and y axes.

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We call these two directions
our coordinate basis

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because they're used to describe 
everything we graph.

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Let's say the tent starts at the origin
and ends up over here at point B.

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The straight arrow connecting
the two points

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is the vector from the origin to B.

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When your friend thinks about
where he has to move,

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it can be written mathematically
as 2x + 3y,

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or, like this, which is called an array.

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Since you're facing the other way,

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your coordinate basis
points in opposite directions,

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which we can call x prime
and y prime,

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and your movement 
can be written like this,

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or with this array.

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If we look at the two arrays,
they're clearly not the same,

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but an array alone doesn't completely
describe a vector.

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Each needs a basis to give it context,

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and when we properly assign them,

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we see that they are in fact
describing the same vector.

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You can think of elements in the array
as individual letters.

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Just as a sequence of letters
only becomes a word

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in the context of a particular language,

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an array acquires meaning as a vector
when assigned a coordinate basis.

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And just as different words
in two languages can convey the same idea,

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different representations from two bases
can describe the same vector.

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The vector is the essence 
of what's being communicated,

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regardless of the language 
used to describe it.

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It turns out that scalars also share
this coordinate invariance property.

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In fact, all quantities with this property
are members of a group called tensors.

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Various types of tensors contain different
amounts of information.

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Does that mean there's something that
can convey more information than vectors?

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Absolutely.

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Say you're designing a video game,

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and you want to realistically model
how water behaves.

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Even if you have forces acting
in the same direction

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with the same magnitude,

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depending on how they're oriented,
you might see waves or whirls.

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When force, a vector, is combined with
another vector that provides orientation,

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we have the physical quantity
called stress,

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which is an example 
of a second order tensor.

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These tensors are also used outside of
video games for all sorts of purposes,

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including scientific simulations,

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car designs,

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and brain imaging.

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Scalars, vectors, and the tensor family
present us with a relatively simple way

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of making sense of complex ideas
and interactions,

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and as such, they're a prime example of
the elegance, beauty,

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and fundamental usefulness of mathematics.