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This online calculator performs vector addition and displays vectors and vector sum graphically.

### This page exists due to the efforts of the following people:

#### Timur

Below you can find vector addition calculator. It calculates the vector sum every time you add an entry into vectors table and also displays results graphically. I've tried to make it as universal as possible, thus you can add vector using two alternative notations - cartesian coordinates (see Cartesian coordinate system) and polar coordinates (see Polar coordinate system). If you choose cartesian, you need to enter x and y components (or coordinates) of a vector. If you choose polar, you need to enter radial (often called the magnitude) and angular (often called the polar angle) components (or coordinates) of a vector. Note that angular coordinate can be entered either as degrees or as radians. Additional details regarding how an addition is performed and how to perform a subtraction can be found below the calculator

#### Vectors

arrow_upwardarrow_downwardCoordinate systemarrow_upwardarrow_downwardX coordinatearrow_upwardarrow_downwardY coordinatearrow_upwardarrow_downwardRadial coordinatearrow_upwardarrow_downwardAngular coordinatearrow_upwardarrow_downwardUnits
Items per page:

Digits after the decimal point: 2

#### Vector sum

X coordinate

Y coordinate

Angular coordinate (degrees)

Internally, calculator converts all entered vectors into cartesian form. It calculates their x and y coordinates using the following conversion formulas:
$x=r cos \theta\\y=r sin \theta$

Then it performs the vector addition, which is very simple and where the vector sum can be expressed as follows:

For vectors $A=(x_1, y_1)$ and $B=(x_2, y_2)$ the vector sum is $A+B=(x_1 + x_2, y_1 + y_2)$

All entered vectors and their sum are also plotted on graph below the results, so you can see the graphical result of the operation, where the vector sum is shown in red. The vector sum is plotted by placing vectors head to tail and drawing the vector from the free tail to the free head (so-called Parallelogram law).

And of course you can use this calculator to calculate vector difference as well, that is, the result of subtracting one vector from another. This is due to the fact, that the vector difference is a vector sum with the second vector reversed, according to:
$A-B=A+(-B)$

To get reversed, or opposite vector in cartesian form, you simply negate the coordinates. In polar form you can either add 180 degrees to angular coordinate, or negate the radial coordinate (either method should work).

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