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It is natural to understand the need for scalars (numbers), but why did we invent vectors? Who invented it and for what?

EDIT: As George Lowther pointed out, the problem is too broad; I added the following questions as concrete supplements.

  1. It's easy for humans to understand the law of addition of scalar numbers, but why does the vector addition follow the parallelogram rule, and not some other law? (link to physics: How do we know that the addition of forces follows the parallelogram law?)

  2. The length of a 2-dimensional vector is the hypotenuse of the triangle constructed from the two components of the vector and the length of 3-dimensional vector also follows this way. But what about 4-dimensional vectors?

  3. Why do we define dot products of vectors like it is now? Is it because of its physical essence or its equivalence to the law of cosines?

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Wow, this question is so broad! Giving a good answer is difficult, since vectors are so ubiquitous that any single response is unlikely to do it justice. Vectors are extremely important in just about every subject of maths and every area of application. I've no idea who invented them, or if that question even makes sense. Any study of maths above a very low level will make use of vectors, whether or not the term "vector" is explicitly used. –  George Lowther Nov 20 '10 at 13:44
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"How do we know that the addition of forces follow the parallelogram law?" If you have a boat on a river and you have a man on each side pulling the boat, the boat moves in a direction that is consistent with the law of the parallelogram. –  Américo Tavares Nov 20 '10 at 16:43
    
Be careful of falling into the trap of classifying that with which you are familiar as "natural" and that with which you are unfamiliar as "unnatural." The more familiar you become with vectors, the more natural they will seem! Soon, you'll see vectors everywhere around you. –  Michael Joyce Jan 2 '12 at 4:06
    
The modern definitions of vectors and vector spaces are due to Hermann Grassman en.wikipedia.org/wiki/Hermann_Grassmann –  Adam Jan 2 '12 at 6:25
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6 Answers

up vote 15 down vote accepted

Vectors should be thought of, at a first approximation, as "numbers with direction". For physical phenomena which carries a direction, such as velocity and displacement, vectors are immensely useful.

The concept of a number with direction most likely dates to antiquity, as the making of maps and sign-posts already implicitly incorporates the notion. The modern representation of a vector/point in space with an ordered triplet of numbers is often attributed to the advent of analytical geometry due to the philosopher Rene Descartes.

A different notion of vectors also arose with the "discovery" of the complex numbers by Jerome Cardan: the imaginary numbers can be thought of as living on a different direction as the real scalars (so the complex numbers form a real vector space).

Over the past 400 years or so the notion of vector gradually evolved to become what we know today, with contributions from branches of mathematics that developed into modern analysis and algebra. A nice summary of that period of development is available here. See Michael Crowe's book for a fuller description of also the Greek contributions and the influences from the 16th century in this matter.

In short, vectors shouldn't be thought of as being "invented", nor should it be attributed to one person alone.

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I think of vectors as "things you can add together", since the modern applications of vectors are so so so broad: colors, functions, propositional calculus, .... –  isomorphismes Dec 20 '10 at 1:13
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Willie covered a lot of what I wanted to say; however I'd like to make a little historical digression: before we ever had the concept of a vector, there was the quaternion, William Rowan Hamilton's generalization of the usual complex numbers. They proved very convenient for physical applications, and thus the use of quaternions took off. In fact the electromagnetic equations of Maxwell were first couched in quaternion notation.

It was Josiah Gibbs and independently Oliver Heaviside who looked at decomposing the quaternion into a scalar (real) part, and a vector (imaginary) part, and found that the manipulations in this new formulation were "cleaner". Vector analysis took off, and quaternions became less prominent.

I'd say more, but a book has already been written about this matter, so I refer you to it.

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Hmm, apparently Willie linked to the Wiki article for the book; I linked to the Dover reprint on Google Books. –  J. M. Nov 20 '10 at 14:02
    
"In fact the electromagnetic equations of Maxwell were first couched in quaternion notation" -- that goes up there with some of his writings on kinetic theory in the list of Not His Finest Moments. :) –  Willie Wong Nov 20 '10 at 14:31
    
@Willie: I remember trying to read those Maxwell papers and ending up with a damn headache. I chalked it up to my limited mathematical ability, and then one day I picked up Crowe's book... –  J. M. Nov 20 '10 at 14:39
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  1. As with most anything in physics, the addition of forces follows the parallelogram law exactly because it agrees with every experiment mankind has ever made---in other words, ultimately it's an assumption we make based on what we see. But, it's perfectly intuitive that this is how forces should behave: say you're at the origin on the xy-plane and have a force pushing you in the positive x-direction and one pushing you in the positive y-direction, both of equal magnitude. Then it should physically and intuitively clear that the net result will be a force pushing you in the direction of (1,1). If the force in the x-direction was stronger, then the net result would still push you in some "diagonal" direction, but aligned more with the x-axis. This is exactly the parallelogram law.

  2. Same exact thing. The problem is that we can't really picture what a right triangle in 4 dimensions looks like, but it's exactly the same idea.

  3. Of course, there are many possible "dot products" that can be defined on vectors, and in many applications it is indeed useful to use one different from the standard one. But I would say the importance of the standard dot product indeed comes from its relation with the cosine of the angle between them: again if you try to think about this physically, this is a precise measure of how much the two vectors (or forces if you like) are working "with" or "against" each other. Note in particular that the sign of the dot product solely depends on whether or not the angle between your two vectors is less than, equal to, or greater than 90 degrees.

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A case study of a particular application:

Vectors are absolutely necessary for game development, for instance. How is a 3D model, which may be positioned at any point in space and rotated in any direction and is viewed by a camera pointing in an arbitrary direction, actually rendered?

First, the model is stored in some local coordinate system. This is the data that actually fills the file from which the model is loaded. Although the exact contents will differ across different rendering systems, at the very least you will have the coordinates of the vertices relative to some arbitrarily chosen center point. These are viewed as vectors, and this viewpoint is crucial for what comes next.

Now all the models are loaded by the rendering system, but they need to be placed into a global coordinate system. This includes putting them in their actual position in the world, rotating them, etc. This stage is accomplished by transforming each of the vectors in local coordinates according to a world transformation matrix--this matrix translates and rotates the model. [You may be wondering how translations are accomplished in this way. That's because most rendering systems store vertices in four-dimensional homogeneous coordinates, and the transformation matrices are $4\times 4$--it's a worthwhile exercise to see that this allows you to perform affine transformations on $\mathbb{R}^3$.]

Next another matrix transformation is applied to all the vertex data to take into account the position and orientation of the camera.

Finally yet another transformation is applied which projects all the data onto the viewing screen--this is not an orthogonal projection but a perspective projection, so that objects in the distance seem smaller than objects close to the camera.

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1) Sean Carroll claims that the essentials of the vector concept emerged with Galileo Galilei. Galileo observed that if you kick a ball forwards off a building, or drop a ball off a building, that the balls land at the same time.

Therefore horizontal motion (kick) is separable from vertical motion (fall). From there you get the separation of the force into two parts: the 2 components of the vector.

3) Regarding the dot product, it is the most natural way to numerically capture the concept of "angle between two things".

2) The pattern continues like this: $\sqrt{a^2 + b^2 + c^2 + d^2 + e^2 + \ldots}$ for a multi-dimensional right triangle with sides $a, b, c, d, e, \ldots$. Surprising, isn't it?

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If vectors are thought of as triplets of numbers (a,b,c) then why should these triplets be added by the parallelogram law? It all depends what these triplets are being used to represent.

If you start with addition of positive numbers on the number line, say 5+3, this can be seen as laying down two line segments of length 5 and 3 along a line starting at the origin. This can be generalized to line segments in a plane by allowing them to point in different directions.

If you start with two adjoined line segments then you can expand the drawing to a parallelogram and it is due to Euclidean geometry that opposite sides of a parallelogram have the same length and point in the same direction. So the parallelogram gives two different paths to move from the origin to the resulting point which is saying that the paralleogram law is equivalent to the commutative law A+B=B+A.

Even in one dimension, the line segments can be given a direction, so that 5+(-3) means putting down a line of length 5 in the +direction, and then turning around and drawing a line of length 3 in the opposite direction to arrive at the number 2. Being commutative you could also start with a line of length 3 in the negative direction from the origin along the real number line and then turn around and draw a line of length 5 in the positive direction again arriving at 2.

So the parallelogram law of vector addition is a straightforward extension of addition of ordinary numbers as directed line segments and is compatible with addition of negative numbers.

Long before cartesian coordinates were developed, people would have used geometry including the pythagorean theorem to calculate the addition of distances and velocities.

Then special relativity was discovered and it was found that the parallelogram law no longer holds for velocity addition, so to answer your question: other laws are indeed sometimes used. The question is: how do we know which laws apply to which aspects of nature? We don't know. We make observations and then make assumptions about how things behave based on those observations. We try to use mathematical structures that accurately describe the observed behaviour, and assume that if our models are accurate enough then any calculations derived from those models will also describe nature accurately.

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