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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,[br]and why do they matter?

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To answer,[br]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 [br]of something there is.

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The distance between you and a bench,

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and the volume and temperature[br]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[br]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[br]and in what direction,

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not just the distance,[br]but the displacement.

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What makes vectors special[br]and useful in all sorts of fields

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is that they don't change [br]based on perspective

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but remain invariant [br]to the coordinate system.

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What does that mean?

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

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

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Your friend moves two steps to the right[br]and three steps forward

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

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

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you both end up moving [br]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[br]over the camp ground,

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the vector doesn't change.

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Let's use the familiar [br]Cartesian coordinate system

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with its x and y axes.

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We call these two directions[br]our coordinate basis

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because they're used to describe [br]everything we graph.

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

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The straight arrow connecting[br]the two points

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is the vector from the origin to B.

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When your friend thinks about[br]where he has to move,

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it can be written mathematically[br]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[br]points in opposite directions,

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which we can call x prime[br]and y prime,

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and your movement [br]can be written like this,

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or with this array.

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

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but an array alone doesn't completely[br]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[br]describing the same vector.

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

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Just as a sequence of letters[br]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[br]when assigned a coordinate basis.

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

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different representations from two bases[br]can describe the same vector.

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

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regardless of the language [br]used to describe it.

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

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

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Various types of tensors contain different[br]amounts of information.

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Does that mean there's something that[br]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[br]how water behaves.

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Even if you have forces acting[br]in the same direction

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with the same magnitude,

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

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

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we have the physical quantity[br]called stress,

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which is an example [br]of a second order tensor.

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These tensors are also used outside of[br]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[br]present us with a relatively simple way

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of making sense of complex ideas[br]and interactions,

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

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and fundamental usefulness of mathematics.