Quick Explanation of Vectors

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Renegade

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Posted 22 May 2012 - 03:29 PM (#1)

Quick Explanation of Vectors

So it took me a long time to grasp the concept of vectors, which in hindsight is silly considering how simple they are when you look at it correctly. So this is a quick explanation of vectors which are critical to any sort of game development or graphics work in general.

What are vectors?
Vectors are quantities which have magnitude and direction. Euclidean Vectors, the common computed type, are essentially right angled triangles which each have a hypotenuse, opposite and adjacent. They are represented with x and y components (and z in 3D space). Let's take an example V(3, 4). Looks like any old geometric point right? That's true however, hidden inside that is direction.

Posted Image

From the above diagram we can see a triangle forming. We can also see that the adjacent i.e. x component is 3 and the opposite i.e. the y component is 4. This is how vectors should be viewed and not as points (which is how I originally interpreted them). From that triangle we can deduce the magnitude and direction of V.

The magnitude or hypotenuse is gotten by Pythagoras which everyone should know to be:

(adjacent)^2 + (opposite)^2 = (hypotenuse)^2

And by heaping the square to the other side, we get the hypotenuse:

sqrt((adjacent)^2 + (opposite)^2)) = hypotenuse
Sqrt(3^2 + 4^2) = hypotenuse
Sqrt(25) = hypotenuse
5 = hypotenuse = magnitude

The direction of the vector is the slope of the hypotenuse which can be gotten using tan:

tan(angle) = opposite/adjacent
tan(angle) = 4/3
Angle = tan^-1(4/3) .. Tan inverse
Angle = 53 degrees

Why is any of this important?
Vectors can represent acceleration, velocity and displacement and any sort of force, which is essential to any physics orientated game. It allows for you to calculate paths, trajectories and basically all aspects of physics within code (maybe a slight exaggeration). Hopefully this short explanation may help your understanding of vectors. Forgive the poor diagrams, Omnigraffle on the iPad is good but not perfect.

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Posted 22 May 2012 - 04:15 PM (#2)

Excellent quick explanation! Physics in games are absolutely crucial, if you want your game physics to look realistic and be believable.

Although I should mention, its not wrong to view them as "points" in relation to graphics since a vector graphics file describes lines as a series of points that need to be connected. Redrawing these points is what causes vector files to be infinitely scalable without losing quality.

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Posted 22 May 2012 - 09:21 PM (#3)

Great initiative! +1 for you.

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Posted 22 May 2012 - 09:22 PM (#4)

And they're extremely easy to add. Although multiplication isn't the same, there is a difference between a x b and a * b. Also you can just add another element to vectors since each one is independent. Then the magnitude just needs to be adjusted to include that new element. Such as adding z or time.

Spoiler

//
//  Vector.cpp
//  Physical
//
//  Created by Mohammad El-Abid on 4/29/12.
//  Copyright (c) 2012 ReliableRabbit.Com. All rights reserved.
//

#include "Vector.h"
#include "Math.h"
#include <sstream>

namespace Physical {
    Vector::Vector(long double x, long double y, long double z) {
        xForce = x;
        yForce = y;
        zForce = z;
    }
    
    long double Vector::magnitude() {
        return sqrt(
            (xForce * xForce) +
            (yForce * yForce) +
            (zForce * zForce)
        );
    }
    
    Vector Vector::add(Vector* vectorTwo) {
        long double newX = xForce + vectorTwo->getXForce();
        long double newY = yForce + vectorTwo->getYForce();
        long double newZ = zForce + vectorTwo->getZForce();
        
        return Vector(newX, newY, newZ);
    }
    
    const std::string Vector::toString() {
        std::stringstream ss;
        ss << "Vector(" << xForce << ", " << yForce << ", " << zForce << ")";
        return ss.str();
    }
}

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NeilHanlon

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Posted 22 May 2012 - 10:22 PM (#5)

Vectors are not easy to add.

Well, in physics they aren't. Cause you don't just add them. Net force and all that jazz. >.>

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Posted 22 May 2012 - 11:18 PM (#6)

NeilHanlon, on 22 May 2012 - 10:22 PM, said:

Vectors are not easy to add.

Well, in physics they aren't. Cause you don't just add them. Net force and all that jazz. >.>

That's applying forces not adding vectors. Different things. You can add two forces acting on the same object to get their net force. Or you add all the forces together to get the sum force.

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NeilHanlon

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Posted 23 May 2012 - 06:14 AM (#7)

No, I'm talking about adding vectors. In Physics, Force is a vector, as is Work, Velocity, Acceleration, etc.

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Posted 23 May 2012 - 06:57 AM (#8)

NeilHanlon, on 23 May 2012 - 06:14 AM, said:

No, I'm talking about adding vectors. In Physics, Force is a vector, as is Work, Velocity, Acceleration, etc.

Depends what form you are given to try to add. Summing polar vectors is nowhere near as neat as adding cartesian vectors, but I still wouldn't classify it as difficult. On the other hand, multiplying polars is far nicer than multiplying cartesians. Regardless though, both forms represent the same conceptual vector, so the statement that vectors are easy to add holds provided you can do the conversion.

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Posted 23 May 2012 - 08:10 AM (#9)

They have quite a large role in Computer Graphics too.

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Posted 23 May 2012 - 12:31 PM (#10)

NeilHanlon, on 23 May 2012 - 06:14 AM, said:

No, I'm talking about adding vectors. In Physics, Force is a vector, as is Work, Velocity, Acceleration, etc.

Adding vectors is not hard, you break it into components and add.

$R_x = A_x + B_x$
$R_y = A_y + B_y$
$R_z = A_z + B_z$

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Posted 23 May 2012 - 12:53 PM (#11)

NeilHanlon, on 23 May 2012 - 06:14 AM, said:

No, I'm talking about adding vectors. In Physics, Force is a vector, as is Work, Velocity, Acceleration, etc.

Actually, Work is energy, so it's a scalar quantity.

Also, as Rails said, adding vectors is easy. In Linear Algebra, vectors are represented by n x 1 matrices, where n is the number of dimensions. Adding two vectors is as simple as adding the components.

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Posted 23 May 2012 - 01:37 PM (#12)

Matrices are funner

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Posted 23 May 2012 - 02:38 PM (#13)

ianonavy, on 23 May 2012 - 12:53 PM, said:

Actually, Work is energy, so it's a scalar quantity.

Actually, actually, work is the change in energy

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Posted 23 May 2012 - 05:50 PM (#14)

ianonavy, on 23 May 2012 - 12:53 PM, said:

Actually, Work is energy, so it's a scalar quantity.

Work is Force over a distance, and the sum of two vectors is a scalar - sorry. Didn't mean to write Work as a vector.

Regardless, whether y'all think adding them is easy, I tend to think it's not. Maybe I just have a teacher who gives us harder problems to do. Who knows. It's my opinion on whether they're easy or hard to add, and you're entitled to your opinion as well. Just don't try to tell me my opinion is wrong...

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Posted 23 May 2012 - 06:46 PM (#15)

NeilHanlon, on 23 May 2012 - 05:50 PM, said:

It is wrong because you are too confused.

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Posted 23 May 2012 - 07:06 PM (#16)

NeilHanlon, on 23 May 2012 - 05:50 PM, said:

The sum of two vectors is definitely still a vector, even if the result is a zero-vector. Did you perhaps mean the scalar (dot) product, as that is analogous to the projection of one vector onto another (with extra cofactors for non-normalised vectors) and hence results in a scalar?

Quote

Force as a vector over a scalar distance would result in a vector.

If you're going to consider force as a vector you need also to consider displacement as existing within the same vector space, hence you use the scalar product from above to get derive a scalar work from the product of the two vectors.

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