A question about vectors, their real life applications and examples, anybody?

[PsychoRebellious]

I was wondering if there are some conditions that imply whether two vectors should be added or multiplied and if rather the multiplication should be dot or cross.
An airplane flies 200km north then 300 km east, find the resultant: in such it is pretty obvious that the resultant will be found by adding them(I dont know why but its just the instinct maybe) BUT why wasn't the resultant found by multipling the two vectors( A=200km north, B=300 km east)

Second question. how can the product of two vectors quantities be a scalar quantity? Yes, we can prove it by dimensional analysis but how would you explain it to somebody who's totally naive? With real life examples, how would two vectors multiplied form a scalar ?

Third: How do you know(Again in real life and not in some dumb exercise questions that say find cross/dot product) when to apply dot product or cross product?

[HTH]

Wat.

Ok So when you add vectors you add them tip to tail.
    300
^------->
|
| 200
|

I cant draw the hypotenuse with ascii but you can see how that works..

If the vectors are at an angle you must take the x,y,z components of it then add them up like
resultant=(x1+x2,y1+y2,z1+z2)

It's trivial to find the length of the vector then. Using root(x^2+y^2...)

Second; stop calling it multiplication.

It's either a dot product or a cross product.

Neither are multiplication, though they may involve it, the DOT product of a series of vectors is used to find relationships between them. Like if they are perpendicular, etc. The CROSS product of two vectors is equal to a third vector which is perpendicular to both Vector1 and Vector2. Subsequently this is only possible in D3 or D5 (IIRC, might work in D9 too but you wont ever see it)

third; see above.

If i said I had a plane that passed through two given points, and gave you their equations, you could find a second point on each (preferably one along the line of intersection) find the vectors between them, and immediately, thanks to dot product, know the relationship between them.

I've taken both linear algebra, which i think youre probably in, and vector calculus. So feel free to message me with any non-retarded questions.

[TheWormKill]

[size=xx-small ! important]1[/size][size=xx-small ! important]1[/size]Vectors are arrows. They have a direction and length, but no position in space.
If you represent movement as a vector and perform multiple movements, you add the corresponding vectors, that's right.
If you multiply a vector with a scalar, you change its length, not direction. Multiplying with a vector with a different vector will result in different things, depending wether you use dot multiplication (the result is a number telling us about the relation between the vectors anglewise) or if you use cross multiplication (the result is a vector that is vertical to both multiplied vectors). But I don't really get what you want to know, please elaborate.

EDIT: HTH was a lot faster. And the terminology I use is shit, I think in a completely different language when it comes to maths.

[Kiuhnm]

I was wondering if there are some conditions that imply whether two vectors should be added or multiplied and if rather the multiplication should be dot or cross.
An airplane flies 200km north then 300 km east, find the resultant: in such it is pretty obvious that the resultant will be found by adding them(I dont know why but its just the instinct maybe) BUT why wasn't the resultant found by multipling the two vectors( A=200km north, B=300 km east)

Second question. how can the product of two vectors quantities be a scalar quantity? Yes, we can prove it by dimensional analysis but how would you explain it to somebody who's totally naive? With real life examples, how would two vectors multiplied form a scalar ?

Third: How do you know(Again in real life and not in some dumb exercise questions that say find cross/dot product) when to apply dot product or cross product?

I'm not sure I understand perfectly your doubts, but, in general, you should know what you want to compute beforehand. Vectors can represent many things and they support many operations. What each operation means depends on the context and on what the vectors represent.
To answer your first question, what does the resultant represent in that context? If you don't have any idea what is the resultant from a qualitative viewpoint then you can't understand why it's computed that way.

As for your second question, the dot product of two vectors is a scalar because that's the way it's defined. The algebraic definition is quite simple:
 a . b = \sum_{i=1}^n a_i b_i
A sum of products is definitely a scalar.
Then you can look at the geometric definition and prove that it's equivalent to the algebraic one.
I can't explain this in real life because math is not about real life. You can use math to explain real life, but math won't tell you if your explanation (aka model) is correct. Using math correctly is your responsibility.
For instance, the math of string theory is correct, but does such theory describe reality? Who knows...

Third question: it depends on what you want to compute. Do you need to compute the angle between two vectors? The normal to a surface? The area of a geometric figure? Etc...

[PsychoRebellious]

Thank you @aboveall for the contributions and answer and I do get what vectors are and what dot and cross products are but my question was more directed towards the applications and usage of these calculations in real world scenarios.
Indeed in a highschool exam there will be questions like "Find the vector and dot product of A and B" but what about the real life?
Just as I gave an example of vector addition scenario that an aeroplane  at A goes 300 km north and reaches a point B , it then travels 400 km east and reaches a point C. The displacement(distance from A - C) can then be calculated by simple adding the components( distance from A to B + distance from B+C) by the head to tail rule. In such a case we have added the two vectors to find the resultant.
Now my question is will there ever be such a scenario when will have to 'multiply' or as corrected by the contributors above 'product'(be that dot or cross) will have to be calculated? One such real life example/application? I don't believe that we are taught about dot and cross product only to solve "Find dot/cross product of A and B" our whole life.
It makes studying distasteful and colorless when you do it for no use, we solve quadratic equations all three years of our student life never knowing when we will be actually using them.

Now about the vector.vector=scalar. @Kiuhnm said it is because just the way it was defined. There must be a reason of it been defined that way? I want to know that reason, I want to know the logic behind it. I can't think of a real life condition where two vectors( say two forces f1 and f2) acting on a body will result in a scalar quantity with no direction involved except in the case of two opposite vectors being added( f1 and f2 are opposite in direction and equal in magnitude)

I've been given the example of
Work=Force*Displacement
Whoever  I asked about this, whichever book I'd consult, it had the above equation how work is a scalar quantity by 'dimensional analysis'
WORK=([N]/[M])*([M])
WORK=[N]

Yes in dimensional analysis the above holds true but lets take the example of an inclined plane. The block on an inclined plane is moved in the direction of the applied force and in this case shouldn't the work be dependent on the direction of Force? If the force was opposite work would be done in opposite direction. Shouldn't work itself be a vector quantity or do we just ignore the direction part of it because we 'want' to?

[HTH]

Ok ill try and break this down for you, but right now im just saving my spot, be patient oh young one and all your questions will be answered.

Thank you @aboveall for the contributions and answer and I do get what vectors are and what dot and cross products are but my question was more directed towards the applications and usage of these calculations in real world scenarios.
Indeed in a highschool exam there will be questions like "Find the vector and dot product of A and B" but what about the real life?
Just as I gave an example of vector addition scenario that an aeroplane  at A goes 300 km north and reaches a point B , it then travels 400 km east and reaches a point C. The displacement(distance from A - C) can then be calculated by simple adding the components( distance from A to B + distance from B+C) by the head to tail rule. In such a case we have added the two vectors to find the resultant.

Well, finding the distance between two or more angled vectors helps IRL, like I know that my house is x km from a spot in the direction of 30* N o E, and y km south from there. I can probably find the actual distance faster than a freshman with a map can.
Using vector addition :p

Now my question is will there ever be such a scenario when will have to 'multiply' or as corrected by the contributors above 'product'(be that dot or cross) will have to be calculated? One such real life example/application? I don't believe that we are taught about dot and cross product only to solve "Find dot/cross product of A and B" our whole life.
It makes studying distasteful and colorless when you do it for no use, we solve quadratic equations all three years of our student life never knowing when we will be actually using them.

You use quadratic equations to solve for X when it is not apparent in an equation with a power of 2 or higher. Honestly if you get fast enough you can find the roots faster that way than somebody doing it by inspection/simplification. Thats a side note though, IRL you probably would never use dot and cross product. Sorry dude. There are some physics questions which you could use it in, like determining the direction a force that needs to be applied to the face of an irregular object on an inclined plane to achieve X m/s^2. But for the most part, dot and cross product are used exclusively in higher level maths. Sorry dude, just the way she goes.

Now about the vector.vector=scalar. @Kiuhnm said it is because just the way it was defined. There must be a reason of it been defined that way? I want to know that reason, I want to know the logic behind it. I can't think of a real life condition where two vectors( say two forces f1 and f2) acting on a body will result in a scalar quantity with no direction involved except in the case of two opposite vectors being added( f1 and f2 are opposite in direction and equal in magnitude)

I mean no offense but this one is easy, vectors are a magnitude and a direction. Let's say that a man is pushing on a wall with a force of 50 N, and an identifcal man join him and applies his force as well at the same spot (doppleganger shit i know) would you not say that the force has been doubled, but the direction stays the same?

tl;dr a vector times a scalar equals a vector with an unchanged direction component because the scalar has no direction component to add to the equation.

EDIT: i see the true meaning of your question
a vector CANNOT be multiplied by a vector. Jesus. the DOT PRODUCT of two vectors is a scalar because it represents the relationship between the two, no more, no less.
x1x2+y1y2+z1z2= dot product, this is used to find the angle between the vectors usually, thats it. the dot product of two vectors isnt a physical quantity, it is unitless and directionless.

Two vectors acting upon an object will NEVER result in just a scalar quantity like the dot product.

I've been given the example of
Work=Force*Displacement
Whoever  I asked about this, whichever book I'd consult, it had the above equation how work is a scalar quantity by 'dimensional analysis'
WORK=([N]/[M])*([M])
WORK=[N]

Yes in dimensional analysis the above holds true but lets take the example of an inclined plane. The block on an inclined plane is moved in the direction of the applied force and in this case shouldn't the work be dependent on the direction of Force? If the force was opposite work would be done in opposite direction. Shouldn't work itself be a vector quantity or do we just ignore the direction part of it because we 'want' to?

work is just that, a measure of work done. Easily enough it is equal to Force times distance. This one is just a 'because joules East" doesnt help us thing. Sorry dude. Work itself could bea vector quantity but we disregard the direction because it's useless. The same amount of work was done no matter if i pushed the block west or east.

[Kiuhnm]

Now about the vector.vector=scalar. @Kiuhnm said it is because just the way it was defined. There must be a reason of it been defined that way? I want to know that reason, I want to know the logic behind it.

Let's say you have a vector space. You can only sum vectors and multiply them by some scalars. If you want to be able to measure norms (lengths) and angles, you need to introduce an inner product. The dot product is the canonical inner product for the Euclidean space.
In general, an inner product is a positive-definite conjugate-symmetric bilinear form. Here are the rules for when the scalars are real numbers (<u,v> indicates the inner product between u and v):
<au,w> = a<u,w>,    a in R
<u+v,w> = <u,w> + <v,w>
<u,v> = <v,u>
<u,u> >= 0 and <u,u> = 0 iff u = 0    [iff = if and only if]

Now let's say we have two vectors u = (a,b) and v = (c,d) in R^2. If e_1 = (1,0) and e_2 = (0,1) we can write them as
u = a e_1 + b e_2
v = c e_1 + d e_2
Therefore, by using the rules above,
<u,v> = <a e_1 + b e_2, c e_1 + d e_2> =
  ac<e_1,e_1> + ad<e_1,e_2> + bc<e_2,e_1> + bd<e_2,e_2>
By convention, <e_i,e_j> = 1 if i=j, and 0 otherwise. So,
<u,v> = ac + bd,
as expected.
BTW, the formula for the determinant of a matrix can be derive in a very similar way.

[Matriplex]

I use vectors ALL the time in game development. They are second nature to me really. Positional vectors and directional vectors both are superbly important in graphics programming. GLSL/HLSL coding, OpenGL coding, DirectX coding all make use of it. I'd say only matrices are more important.

What stinks is I haven't actually taken any math classes that even deal with this stuff. I'm only in algebra II/trig when I could/should be in ap calc. Ah education systems.

[PsychoRebellious]

Let's say you have a vector space. You can only sum vectors and multiply them by some scalars. If you want to be able to measure norms (lengths) and angles, you need to introduce an inner product. The dot product is the canonical inner product for the Euclidean space.
In general, an inner product is a positive-definite conjugate-symmetric bilinear form. Here are the rules for when the scalars are real numbers (<u,v> indicates the inner product between u and v):
<au,w> = a<u,w>,    a in R
<u+v,w> = <u,w> + <v,w>
<u,v> = <v,u>
<u,u> >= 0 and <u,u> = 0 iff u = 0    [iff = if and only if]

Now let's say we have two vectors u = (a,b) and v = (c,d) in R^2. If e_1 = (1,0) and e_2 = (0,1) we can write them as
u = a e_1 + b e_2
v = c e_1 + d e_2
Therefore, by using the rules above,
<u,v> = <a e_1 + b e_2, c e_1 + d e_2> =
  ac<e_1,e_1> + ad<e_1,e_2> + bc<e_2,e_1> + bd<e_2,e_2>
By convention, <e_i,e_j> = 1 if i=j, and 0 otherwise. So,
<u,v> = ac + bd,
as expected.
BTW, the formula for the determinant of a matrix can be derive in a very similar way.

How I wish I could say this in a less embarrassing way but I am so totally lost with all the geometrical complications, the inner product, cannonical inner product, conjugate symmetrical binomial form- Don't even if that's geometry or Aglebra. Times when I wish I wasn't so ignorant. I 'll have to study about all these things before grasping what you actually said there.

@HTH: "Two vectors being multiplied will never give a scalar quantity"- with that all said we can assume dot product is not the same as 'multiplication' of two vectors but some other relation of them? I think I got confused because in this book it was written
quote
"Vectors can be multiplied in two ways, when they form a scalar product its called dot product and when they form a vector product its called a cross product"
Applying even dimensional analyses the 'product' of two vectors lets say displacement*displacement should be displacement^2  or meter^2. I don't know what that'd be, a vector in two dimensions? Aren't all vectors in 2 dimensions with a component in each of them?

Secondly if no direction is involved in dot product then why does the formula go
A.B=ABcosx  ??
I read all about the Bcosx is the projection of B over A. In real life example I can think of an object casting a shadow on the ground but that(the projection) would depend on the distance and position of light source too and who in their buttfucking right mind want to know the product of shadow and the area it covers on ground?
I might be talking real nonsense here but i don't want to move on with the topic only rhyming
A.B=abcosx
AxB=absinx
without knowing what's actually going on.
EDIT: Oops forgot that you said it will be used in high level mathematics and I won't find an application of it at this level. How very comforting :-/

@Matriplex: Game development, yes, I've heard how you need to be good with linear algebra and vector analysis when with 3d game development. Yet to try it out myself. Education system being shitty is undebatable

[HTH]

@HTH: "Two vectors being multiplied will never give a scalar quantity"- with that all said we can assume dot product is not the same as 'multiplication' of two vectors but some other relation of them? I think I got confused because in this book it was written
quote

Yes. as I've tried to say multiple times. Thats why I don't like using the term multiplication, it confuses new people.

"Vectors can be multiplied in two ways, when they form a scalar product its called dot product and when they form a vector product its called a cross product"

That is the dumbest explanation ever.

"There are two 'multiplcation'-style operations that can be performed upon vectors. The first; Dot product, is calculated from two vectors A and B with the formula AxBx+AyBy=Dot product. The second, cross product, well to be quite frank I dont use a formula I use a x by x matrice and solve them quickly using determinants."

Either way your version is retardedly worded.

Applying even dimensional analyses the 'product' of two vectors lets say displacement*displacement should be displacement^2  or meter^2. I don't know what that'd be, a vector in two dimensions? Aren't all vectors in 2 dimensions with a component in each of them?

it is not vector*vectors. stop thinking it is. But yes a vector in a dimension D^a will have a component in each of the a directions. *2 in 2d, 3 in 3d, etc*

Secondly if no direction is involved in dot product then why does the formula go
A.B=ABcosx  ??

Lol well frankly because that isnt the formula, that is the start to the proof of the communicative property of dot products. Where A.B = ABcos(theta) = BAcos(theta) = B.A

I can further simplify it using scalar projection or the geometric def'n of the dot product which utilizes a unit vector with an implicit magnitude of 1.. but id ratehr not explain it to you when you don't know the correct formulas, no offense..

I read all about the Bcosx is the projection of B over A. In real life example I can think of an object casting a shadow on the ground but that(the projection) would depend on the distance and position of light source too and who in their buttfucking right mind want to know the product of shadow and the area it covers on ground?

This part is quite simple lol. The scalar projection of A over B is equal to Acos(theta) because if you make a right triangle then A is the hypotenus and B is the adjacent, and cos=adj/hyp.

You can use it any time you need to find that length :p

I might be talking real nonsense here but i don't want to move on with the topic only rhyming
A.B=abcosx
AxB=absinx
without knowing what's actually going on.
EDIT: Oops forgot that you said it will be used in high level mathematics and I won't find an application of it at this level. How very comforting :-/

Right I'm gonna tell you right now that A.B might equal ABcosx but that is NOT the formula that you should use. That formula is used in proofs and shouldnt be taught at all at your level lol, it should have been a quick overview if anything just saying "yeah this proves A.B = B.A"
same for the cross product, thats the geometric definition, NOT the formula to find it.

A.B = Ax*Bx + Ay*By
or, the sum of the products of the components.

and the cross product is equal to... ah fuck it.

(x,y,z) X (x2,y2,z2) = (xc,yc,zc)
[ xc   yc   zc  ]
[ x    y     z   ]
[ x2  y2   z2 ]

where
xc = (y*z2) - (z*y2)
yc = (x*z2) - (z*x2)
zc = (x*y2) - (y-x2)

This can be remembered easily as taking the deteminant
[  a  b ]
[  c  d ]
det = ad-cb

in relation to the points that you wish to find, so for xc you disregard the row and column that it is in, same goes for yc, and zc.

Im sorry but outside of math you wont find an application for this any time soon, at least not a physical one. If you need any math help, again, ill usually respond to PM's. But I think you're either over thinking this or blessed with a very shitty teacher.

[Kiuhnm]

How I wish I could say this in a less embarrassing way but I am so totally lost with all the geometrical complications, the inner product, cannonical inner product, conjugate symmetrical binomial form- Don't even if that's geometry or Aglebra. Times when I wish I wasn't so ignorant. I 'll have to study about all these things before grasping what you actually said there.

That's linear algebra.

@HTH: "Two vectors being multiplied will never give a scalar quantity"- with that all said we can assume dot product is not the same as 'multiplication' of two vectors but some other relation of them? I think I got confused because in this book it was written
quote
"Vectors can be multiplied in two ways, when they form a scalar product its called dot product and when they form a vector product its called a cross product"
Applying even dimensional analyses the 'product' of two vectors lets say displacement*displacement should be displacement^2  or meter^2. I don't know what that'd be, a vector in two dimensions? Aren't all vectors in 2 dimensions with a component in each of them?

A product is the result of a multiplication, so I see nothing wrong with speaking of multiplication between vectors.
But you're making a conceptual mistake there: the product of two vectors is NOT the product of two real numbers. Stop mixing things up.
Having said that, the dot product is called a 'product' because it has some properties in common with the product between numbers. For instance,
  x . (y + z) = x.y + x.z
Looks familiar?

Secondly if no direction is involved in dot product then why does the formula go
A.B=ABcosx  ??

What does it mean "no direction is involved in dot product"?

I might be talking real nonsense here but i don't want to move on with the topic only rhyming
A.B=abcosx
AxB=absinx
without knowing what's actually going on.
EDIT: Oops forgot that you said it will be used in high level mathematics and I won't find an application of it at this level. How very comforting :-/

The second formula is wrong: AxB is a vector, not a scalar. It should be
AxB = ||A|| ||B|| sinx n
where n is orthogonal to A and B.

The sooner you make peace with the fact that there's nothing going on, the better
IMHO, you should learn how things are and leave the philosophical questions for when you're more knowledgeable.

[Kiuhnm]

Lol well frankly because that isnt the formula, that is the start to the proof of the communicative property of dot products. Where A.B = ABcos(theta) = BAcos(theta) = B.A

There's no need to drag in that formula:
A.B = \sum_{i=1}^n A_i B_i
      = \sum_{i=1}^n B_i A_i
      = B.A

[PsychoRebellious]

Honestly speaking I haven't yet stumbled upon the formula @hth mentioned

Right I'm gonna tell you right now that A.B might equal ABcosx but that is NOT the formula that you should use. That formula is used in proofs and shouldnt be taught at all at your level lol, it should have been a quick overview if anything just saying "yeah this proves A.B = B.A"
same for the cross product, thats the geometric definition, NOT the formula to find it.

A.B = Ax*Bx + Ay*By
or, the sum of the products of the components.

Here I am going to copy paste a paragraph from the book "Fundamental of physics by resnick hallidayd and crane"

Multiplying a Vector by a Vector:
There are two ways to multiply a vector by a vector: one way produces a scalar
(called the scalar product), and the other produces a new vector (called the vector
product). (Students commonly confuse the two ways.)
The Scalar Product:
The scalar product of the vectors A and B and in Fig. 3-18a is written as A.B and
defined to be
A.B=ab cos theeta

where a is the magnitude of A, b is the magnitude of B, and theeta is the angle between
A and B

Vector product:
The vector product of A and B, written AXB , produces a third vector C whose
magnitude is
c=ab sintheeta,

Its the only formula that we are taught in highschool, Kudos!

EDIT:
This all makes me realize how I should look more on subject. I asked these questions from a few friends of mine who had already given and passed highschool exams with flying colors, none of them could answer those. Thank you all for the answers. Many of my doubts are cleared by your efforts 
Edit again:
Oh and about the 'having a bad teacher' part then no, don't have a teacher, let alone a bad one so I have to strive with online communities and google. Failed my highschool exams once. Found the subject boring. My parents have sort of given up on me, they are not going to waste money on my tutiouns or stuff so I here I am with these books that do not make sense and my exams(that I am going to give privately because thats the cheapest option available) in the beginning of december and I'm only on the secondly chapter of hSC-I (vectors). I have the whole HSC-I and HSC-II to cover :')

[Kiuhnm]

If you're in high school then forget about inner products and bilinear forms. That's university-level stuff. My bad.
If I were you, I'd use a book with many exercises (also worked out) placed inside the chapters and not just at the end. Then you can try the exercises and ask for help when you get stuck.
At least that's the way I like to learn math (I'm not a physicist).

[HTH]

AxB = ||A|| ||B|| sinx n
where n is orthogonal to A and B.

Actually your format is more incorrect than his.
AxB = ||A|| ||B|| sinx is correct, which is why I didn't comment on the lack of ||'s

AxB is orthogonal to A and B, not some made up variable n at the end of the formula.

Either way though both of the formulae that have trig in them are useless in actually solving things.

There's no need to drag in that formula:
A.B = \sum_{i=1}^n A_i B_i
      = \sum_{i=1}^n B_i A_i
      = B.A

he brought it up to start with I was explaining where it came from.Nobody uses
A.B = abcosx
or
AxB = absinx

except in proofs, and that's up to Vector Calculus(commonly a third year math here but i took it in second) and Third Year Physics classes in University. SO maybe after that, in a graduate program? people would use them on a day to day .... I already mentioned the two formulae you actually need. I'm gonna bow out of this thread now though, I've already said everything I can on the subject (well not really) but as a tl;dr

Vector 'multiplication' isn't the same as multiplication of scalar quantities. You can use the dot product with the proof (geometric) formula of A.B = ABcos(theta) and the REAL formula of A.B = AxBx + AyBy. Or you can use the cross product with the proof (geometric) formula of AxB = ABsin(theta) and the real working formula i mentioned above. (using three separate determinates).

Either way I hope you figure this out. GLHF