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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 the 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.