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Cartesian Coordinates
3D Cartesian Coordinates
In the simulation below you can see the cartesian coordinate system measuring the distance between the head of the arrow and its ending point, starting at the origin, (0,0,0). Notice how there are x, y, and z vector components accompanying the vector. In the 3D Cartesian coordinate system, components are useful when determining the distance length between three points. Using the Pythagorean Theorem, the distance between three points can be easily determined.
Cartesian coordinates are a coordinate system that uses the x, y, and z axis on a two dimensional plane, and an additional z axis for a three dimensional area. Cartesian coordinates, also known as rectangular coordinates, are best used to measure the distances between points. For example, you can measure points: away from the origin (0,0,0), create a polygon shape such as a triangle, rectangle, etc, or perform transformation operations such as reflection, translation, rotation, or dilation. It is worth noting that in Cartesian Coordinates, the x, y, and z axes range from 0 to ∞ in the positive direction and from 0 to -∞ in the negative direction. The cartesian coordinate system differs from the polar coordinate system in that the x, y, and z components of an object are provided, and the distance and angles are left to be calculated.
The simulation above shows the cartesian coordinate system used in a three dimensional area, with the x, y, and z axes. The simulation above best represents determining the distance of three points. For example if you enter the points: 3 in the x axis, 4 in the y axis, and 5 in the z axis, you will obtain a distance of 7.071 between the three points. Similarly, if you enter 0 in the x axis, 13 in the y axis, and 7 in the z axis, you will obtain a distance of 14.765 between the three points. If you enter any number in the the simulation, for example entering 2 for x axis, 3 for y axis, and 6 for the z axis, you will obtain a distance of 7 between the three points, in this case, this can be calculated using the Pythagorean Theorem.
Pythagorean Theorem:
\[ a^2 + b^2+ c^2 = d^2 \]
This is how to calculate the distance 17, of the vector starting from (0, 0, 0) and ending at (2, 3, 6):
\[ a^2 + b^2 + c^2 = d^2 \]
\[ 2^2 + 3^2 + 6^2 = d^2 \]
\[ 4 + 9 + 36 = d^2 \]
\[ 49 = d^2 \]
\[ \sqrt{49} = \sqrt{d^2} \]
\[ 7 = d \]
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