What is a vector? - David Huynh

Prevodilac: Ema Maričić Lektor: Tijana Mihajlović

Physicists,

Fizičari,

kontrolori leta

air traffic controllers,

and video game creators

i kreatori video-igara

all have at least one thing in common:

imaju bar jednu zajedničku stvar:

vectors.

vektore.

What exactly are they, and why do they matter?

Šta su oni zaista i zašto su važni?

To answer, we first need to understand scalars.

Da bismo odgovorili na to, moramo prvo da razumemo skalare.

A scalar is a quantity with magnitude.

Skalar je kvantitet sa brojnom veličinom.

It tells us how much of something there is.

Govori nam koliko ima nečega.

The distance between you and a bench,

Razdaljina između vas i klupe,

and the volume and temperature of the beverage in your cup

zapremina i temperatura pića u vašoj šolji

are all described by scalars.

opisani su skalarima.

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

Vektori takođe imaju veličinu, kao i dodatnu informaciju,

direction.

pravac.

To navigate to your bench,

Da biste stigli do klupe,

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

trebalo bi da znate koliko je udaljena i u kom pravcu je;

not just the distance, but the displacement.

a ne samo udaljenost, već i premeštanje u drugi položaj.

Ono što čini vektore posebnim i upotrebljivim u različitim poljima

What makes vectors special and useful in all sorts of fields

is that they don't change based on perspective

je što se ne menjaju u zavisnosti od perspektive,

but remain invariant to the coordinate system.

već ostaju nepromenljivi u odnosu na koordinantni sistem.

What does that mean?

Šta to znači?

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

Recimo da vi i vaš prijatelj premeštate šator.

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

Stojite na različitim stranama i gledate u suprotnim smerovima.

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

Vaš prijatelj se pomeri dva koraka udesno i tri napred,

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

a vi se pomerite dva koraka levo i tri nazad.

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

Iako izgleda kao da se krećete različito,

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

oboje se prelazite istu udaljenost u istom smeru,

following the same vector.

prateći isti vektor.

No matter which way you face,

Bez obzira na to na koju stranu gledate

or what coordinate system you place over the camp ground,

ili koji koordinantni sistem stavite na izletište,

the vector doesn't change.

vektor se ne menja.

Let's use the familiar Cartesian coordinate system

Uzmimo Dekartov koordinantni sistem,

with its x and y axes.

sa osama x i y.

We call these two directions our coordinate basis

Ta dva pravca nazivamo koordinantim osama

because they're used to describe everything we graph.

jer se koriste za opisivanje svega što predstavimo grafikonom.

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

Recimo da se šator kreće od početka do tačke B.

The straight arrow connecting the two points

Prava strelica koja povezuje ove dve tačke

is the vector from the origin to B.

je vektor od mesta polaska do tačke B.

When your friend thinks about where he has to move,

Kad vaš prijatelj razmišlja kako treba da se pomeri,

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

to se može matematički opisati kao 2x + 3y,

or, like this, which is called an array.

ili ovako, što nazivamo nizom.

Since you're facing the other way,

Kako gledate u suprotnim smerovima,

your coordinate basis points in opposite directions,

vaše koordinatne ose su suprotno okrenute

which we can call x prime and y prime,

i njih ćemo nazvati x' i y',

and your movement can be written like this,

a vaše kretanje se može ovako pribeležiti

or with this array.

ili ovim nizom.

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

Ako pogledamo ta dva niza, videćemo da definitivno nisu isti,

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

ali sam niz ne opisuje vektor u potpunosti.

Each needs a basis to give it context,

Svaki zahteva osu da bi imao kontekst,

and when we properly assign them,

a kad ih pravilno pripišemo,

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

možemo videti da zapravo opisuju isti vektor.

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

Možete da gledate na elemente niza kao na pojedinačna slova.

Just as a sequence of letters only becomes a word

Kao što niz slova postaje reč

in the context of a particular language,

samo kada je u kontekstu u nekom jeziku,

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

tako i niz dobija značenje kao vektor kad mu se pripiše koordinantna osa.

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

I kao što različite reči u dva jezika mogu da govore o istom pojmu,

different representations from two bases can describe the same vector.

različite predstave dve ose opisuju isti vektor.

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

Vektor je ključan za ono o čemu se priča,

regardless of the language used to describe it.

bez obzira na to koji jezik se koristi za opisivanje.

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

Ispostavlja se da su skalari konstantni u odnosu na koordinantni sistem.

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

Zapravo, svi kvantiteti sa ovom osobinom su u skupu tenzora.

Various types of tensors contain different amounts of information.

Različiti tipovi tenzora sadrže različite količine informacija.

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

Da li to znači da postoji nešto što može da prenosi više informacija od vektora?

Absolutely.

Naravno.

Say you're designing a video game,

Recimo da dizajnirate video-igru

and you want to realistically model how water behaves.

i želite da realistično modelirate kretanje vode.

Even if you have forces acting in the same direction

Čak i ako imate sile koje deluju u istom pravcu

with the same magnitude,

istim intenzitetom,

u zavisnosti od njihove orijentacije, možete videti talasanje ili kovitlanje.

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

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

Kada se sila, vektor, kombinuje sa drugim vektorom koji pruža orijentaciju,

we have the physical quantity called stress,

onda se dobija fizički kvantitet poznat kao napon,

which is an example of a second order tensor.

a on je primer tenzora drugog reda.

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

Ovi tenzori se koriste ne samo za video-igre, već imaju različite namene,

including scientific simulations,

poput naučnih simulacija,

car designs,

dizajniranja automobila

and brain imaging.

i snimanja mozga.

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

Skalari, vektori i porodica tenzora pružaju nam relativno lak način

of making sense of complex ideas and interactions,

razjašnjavanja kompleksnih ideja i interakcija

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

i, kao takvi, primer su elegancije, lepote

and fundamental usefulness of mathematics.

i fundamentalne primenljivosti matematike.