What is a vector? - David Huynh
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0:07 - 0:08Physicists,
-
0:08 - 0:10air traffic controllers,
-
0:10 - 0:11and video game creators
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0:11 - 0:14all have at least one thing in common:
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0:14 - 0:16vectors.
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0:16 - 0:19What exactly are they,
and why do they matter? -
0:19 - 0:23To answer,
we first need to understand scalars. -
0:23 - 0:26A scalar is a quantity with magnitude.
-
0:26 - 0:29It tells us how much
of something there is. -
0:29 - 0:31The distance between you and a bench,
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0:31 - 0:35and the volume and temperature
of the beverage in your cup -
0:35 - 0:38are all described by scalars.
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0:38 - 0:43Vector quantities also have a magnitude
plus an extra piece of information, -
0:43 - 0:44direction.
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0:44 - 0:46To navigate to your bench,
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0:46 - 0:50you need to know how far away it is
and in what direction, -
0:50 - 0:53not just the distance,
but the displacement. -
0:53 - 0:57What makes vectors special
and useful in all sorts of fields -
0:57 - 1:00is that they don't change
based on perspective -
1:00 - 1:03but remain invariant
to the coordinate system. -
1:03 - 1:05What does that mean?
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1:05 - 1:08Let's say you and a friend
are moving your tent. -
1:08 - 1:12You stand on opposite sides
so you're facing in opposite directions. -
1:12 - 1:16Your friend moves two steps to the right
and three steps forward -
1:16 - 1:19while you move two steps to the left
and three steps back. -
1:19 - 1:22But even though it seems
like you're moving differently, -
1:22 - 1:26you both end up moving
the same distance in the same direction -
1:26 - 1:28following the same vector.
-
1:28 - 1:30No matter which way you face,
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1:30 - 1:33or what coordinate system you place
over the camp ground, -
1:33 - 1:36the vector doesn't change.
-
1:36 - 1:38Let's use the familiar
Cartesian coordinate system -
1:38 - 1:41with its x and y axes.
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1:41 - 1:44We call these two directions
our coordinate basis -
1:44 - 1:47because they're used to describe
everything we graph. -
1:47 - 1:52Let's say the tent starts at the origin
and ends up over here at point B. -
1:52 - 1:54The straight arrow connecting
the two points -
1:54 - 1:57is the vector from the origin to B.
-
1:57 - 2:00When your friend thinks about
where he has to move, -
2:00 - 2:04it can be written mathematically
as 2x + 3y, -
2:04 - 2:07or, like this, which is called an array.
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2:07 - 2:09Since you're facing the other way,
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2:09 - 2:12your coordinate basis
points in opposite directions, -
2:12 - 2:15which we can call x prime
and y prime, -
2:15 - 2:19and your movement
can be written like this, -
2:19 - 2:22or with this array.
-
2:22 - 2:25If we look at the two arrays,
they're clearly not the same, -
2:25 - 2:30but an array alone doesn't completely
describe a vector. -
2:30 - 2:33Each needs a basis to give it context,
-
2:33 - 2:34and when we properly assign them,
-
2:34 - 2:38we see that they are in fact
describing the same vector. -
2:38 - 2:42You can think of elements in the array
as individual letters. -
2:42 - 2:45Just as a sequence of letters
only becomes a word -
2:45 - 2:48in the context of a particular language,
-
2:48 - 2:53an array acquires meaning as a vector
when assigned a coordinate basis. -
2:53 - 2:57And just as different words
in two languages can convey the same idea, -
2:57 - 3:02different representations from two bases
can describe the same vector. -
3:02 - 3:05The vector is the essence
of what's being communicated, -
3:05 - 3:08regardless of the language
used to describe it. -
3:08 - 3:13It turns out that scalars also share
this coordinate invariance property. -
3:13 - 3:18In fact, all quantities with this property
are members of a group called tensors. -
3:18 - 3:23Various types of tensors contain different
amounts of information. -
3:23 - 3:27Does that mean there's something that
can convey more information than vectors? -
3:27 - 3:28Absolutely.
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3:28 - 3:30Say you're designing a video game,
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3:30 - 3:34and you want to realistically model
how water behaves. -
3:34 - 3:37Even if you have forces acting
in the same direction -
3:37 - 3:38with the same magnitude,
-
3:38 - 3:43depending on how they're oriented,
you might see waves or whirls. -
3:43 - 3:48When force, a vector, is combined with
another vector that provides orientation, -
3:48 - 3:51we have the physical quantity
called stress, -
3:51 - 3:54which is an example
of a second order tensor. -
3:54 - 4:00These tensors are also used outside of
video games for all sorts of purposes, -
4:00 - 4:01including scientific simulations,
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4:01 - 4:03car designs,
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4:03 - 4:04and brain imaging.
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4:04 - 4:09Scalars, vectors, and the tensor family
present us with a relatively simple way -
4:09 - 4:13of making sense of complex ideas
and interactions, -
4:13 - 4:17and as such, they're a prime example of
the elegance, beauty, -
4:17 - 4:20and fundamental usefulness of mathematics.
- Title:
- What is a vector? - David Huynh
- Description:
-
View full lesson: http://ed.ted.com/lessons/what-is-a-vector-david-huynh
Physicists, air traffic controllers, and video game creators all have at least one thing in common: vectors. But what exactly are they, and why do they matter? David Huynh explains how vectors are a prime example of the elegance, beauty, and fundamental usefulness of mathematics.
Lesson by David Huynh, animation by Anton Trofimov.
- Video Language:
- English
- Team:
- closed TED
- Project:
- TED-Ed
- Duration:
- 04:41
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