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Cartesian Coordinates

2D 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). Notice how there are x and y vector components accompanying the vector. In the 2D Cartesian coordinate system, components are useful when determining the distance length between two points. Using the Pythagorean Theorem, the distance between two points can be easily determined.

Cartesian coordinates are a coordinate system that uses the x and y 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), 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 and y 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 and y 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 two dimensional plane, with the x and y axes. The simulation above best represents determining the distance of two points. For example if you move your mouse to the points 23 in the x axis and 0 in the y axis you will obtain a distance of 23 between the two points. Similarly, if you move your mouse to 0 in the x axis and 14 in the y axis you will obtain a distance of 14 between the two points. If you move your mouse anywhere on the grid, for example over 8 in the x axis and 15 in the y axis, you will obtain a distance of 17 between the two points, in this case, this can be calculated using the Pythagorean Theorem.

Pythagorean Theorem:

\[ a^2 + b^2 = c^2 \]

This is how to calculate the distance 17, of the vector starting from (0, 0) and ending at (8, 15):

\[ a^2 + b^2 = c^2 \]

\[ 8^2 + 15^2 = c^2 \]

\[ 64 + 225 = c^2 \]

\[ 289 = c^2 \]

\[ \sqrt{289} = \sqrt{c^2} \]

\[ 17 = c \]

Using this simulation you can also perform most transformations:

1. Reflection:

To perform a reflection we will start with an arbitrary point, with moving your mouse over 12 in the x axis, and over 9 in the y axis. This will result in a length between (0, 0) and (12, 9) to be 15. To reflect this line (vector) over the x axis, you will move your mouse to 12 in the x axis, and over -9 in the y axis. Notice that the length between (0, 0) and (12, -9) is still 15, the difference here however is that this new line (vector) is reflected over the x axis, resulting in a new direction and orientation of the line (vector). Similarly to reflect this line (vector) over the y axis, you will move your mouse to -12 in the x axis, and over 9 in the y axis. Notice that the length between (0, 0) and (-12, 9) is still 15, the difference here however is that this new line (vector) is reflected over the y axis, resulting in a new direction and orientation of the line (vector).

Reflection over the x axis is following this rule (x, y) → (x, -y):

\[ (x, y) = (x, -y) \]

\[ (12, 9) = (12, -9) \]

Reflection over the y axis is following this rule (x, y) → (-x, y):

\[ (x, y) = (-x, y) \]

\[ (12, 9) = (-12, 9) \]

Reflection over y = x is following this rule (x, y) → (y, x):

\[ (x, y) = (y, x) \]

\[ (12, 9) = (9, 12) \]

Reflection over y = -x is following this rule (x, y) → (-y, -x):

\[ (x, y) = (-y, -x) \]

\[ (12, 9) = (-9, -12) \]

2. Translation:

Now although translation cannot be represented in the simulation, translation relates to shifting an object from its original position to a new position. Let us say a rectangle has the coordinates: (0, 0), (2, 0), (0, 3), (2, 3). You then decide to translate or shift the box 5 units to the right (positive direction), and 4 units down (negative direction). The new coordinates of the rectangle now become: (5, -4), (7, -4), (5, -1), (7, -1).

Translating right (positive x axis) can be performed adding a number k to coordinate points:

\[ (x, y) = (x + k, y) \]

Translating left (negative x axis) can be performed subtracting a number k to coordinate points:

\[ (x, y) = (x - k, y) \]

Translating up (positive y axis) can be performed adding a number k to coordinate points:

\[ (x, y) = (x, y + k) \]

Translating down (negative y axis) can be performed subtracting a number k to coordinate points:

\[ (x, y) = (x, y - k) \]

3. Rotation:

As you move your mouse around the simulation, changing the position of the line (vector), you are rotating the line (vector). The proper definition of rotation is the circular movement of a fixed object. As you move your mouse in the simulation, you are changing both the angle and length of the line (vector). For proper rotation, you only change the angle of the object, but the object must have a fixed length. An example of a 90° rotation using the simulation would be to have the line (vector) start at the coordinates 5 in the x axis and 12 in the y axis, resulting in a length of 13. To rotate this line (vector) by 90 degrees, the x axis coordinate becomes -12, and the y axis coordinate becomes 5. Notice that the distance between the origin (0, 0) and the rotated points does not change, it is still 13. This is applicable to an object rotated at any angle as long as the object's lengths are fixed.

Rotation in 90° follows this rule (x, y) → (-y, x):

\[ (x, y) = (-y, x) \]

\[ (5, 12) = (-12, 5) \]

Rotation in 180° follows this rule (x, y) → (-x, -y):

\[ (x, y) = (-x, -y) \]

\[ (5, 12) = (-5, -12) \]

Rotation in 270° follows this rule (x, y) → (y, -x):

\[ (x, y) = (y, -x) \]

\[ (5, 12) = (12, -5) \]

Rotation in 360° follows this rule (x, y) → (x, y):

\[ (x, y) = (x, y) \]

\[ (5, 12) = (5, 12) \]

4. Dilation:

To perform a dilation, move your mouse over 4 in the x axis, and over 3 in the y axis. This will result in a length between (0, 0) and (4, 3) to be 5. Now move your mouse to the points 12 in the x axis, and over 9 in the y axis. This will result in a length between (0, 0) and (12, 9) to be 15. Notice that you just increased the distance of the line (vector) by a factor of 3, because initially the distance was 5, now it is 15. This is dilation. You only changed the length (size) of the line (vector) however, you have not changed the direction of the vector. This is applicable to polygons as well. You keep the direction and shape of an object, but only change its size by dilating it. In terms of vectors, this is called vector scaling. Although a different name, this is the same idea of changing an object's original size to smaller or larger.

Dilation can be performed multiplying the coordinate points by the dilated factor k:

\[ (x, y) = (kx, ky) \]

\[ k = 3 \]

\[ (4, 3) = ((3)(4), (3)(3)) \]

\[ (4, 3) = (12, 9) \]

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