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Dimensions can be more accurately calculated when using algabreic equations rather than numerical values.
Two or more straight lines meeting at one point.
You need to be a bit more specific in your question. You state you have 1 AC system but 2 thermostats. What type of heating system do you have? Gas furnace, Boiler, Electric heat, Hot Water? Is your heating unit separate from your cooling unit?
The K map does not necessarily "fail" for higher dimensions. The problem is that it is so difficult to visualize for more than five variables. A 4 variable K-map is 2 dimensional and easy to visualize. A 5 variable is three dimensional, but is still manageable from a visualization standpoint, because the 2 states of the 5th variable only require visually moving from one plane to the next, without moving in the x or y directions of either plane. Just getting equations correct with more than 5 variables is difficult enough using the K map, much less considering an optimum set of terms ("core" prime implicants and "choice" prime implicates).
The word complicates is the present tense of the verb to complicate, meaning to make more complicated or more complex, or in other words, to complicate is the opposite of to simplify. The addition of more details, more problems, more aspects of a given situation are all things that can complicate it.
The Cartesian coordinate system can be used in 3 or more dimensions.
Yes, the Cartesian coordinate system is routinely extended to 3 and more dimensions. In 3-d the location of each point is determined by an ordered triple, usually denoted (x,y,z), with corresponding extensions to more dimensions.
The cartesian coordinate system is the basic coordinate system people are familiar with. If you were to plot a basic X Y graph, where X and Y increase towards infinity, you are graphing in the cartesian coordinate system. There are a few other popular coordinate systems for more advanced math, but they are rarely used by most people. (polar coordinate system is common for some calculus equations)
It is a point on the coordinate grid. The grid may be the Cartesian or coordinate plane, or its equivalent in 3 or more dimensions. It could also refer to a grid where the axes are not at right angles to one another (eg isometric grid).
It is called The Origin. You can go to Wikipedia.com and get more information on The Origin.
A "Cartesian Ordered Pair," more commonly known as simply an "Ordered Pair."
I suspect the question arises from confusion. A vector itself already defines a direction, usually in the Cartesian xyz coordinate system. If you want to express the direction in other coordinates, such as polar or spherical coordinates you need to transform the vector to these coordinate systems. I can answer you question more fully if you can specify the specific coordinate system in which you want to know the direction.
Yes, almost all the time. And when it is not coordinate planes it is coordinate hype-spaces (more than just the 2 dimensions that the coordinate plane allows).
Yes. All you need is three mutually perpendicular axes (instead of two). To visualise this, look at the corner of a room. There will be three lines coming together at the corner: floor and one wall, floor and another wall, and the two walls. These three lines would act as your axes to describe the 3-d space of the room. The axes are usually labelled x, y and z. Mathematicians (and physicists) have no problem in dealing with coordinate systems in 4 or more dimensions.
To describe position, you need more than one number - for instance, an x-coordinate, a y-coordinate, and (if it is in three dimensions) a z-coordinate. That's the very essence of a vector - the fact that it is made up of several components.
The name Cartesian is derived from the name Descartes, after its creator, Rene Descartes. In math, a Cartesian plane is the entire field of a graph. The horizontal number line is called the x-axis, and the vertical line is called the y-axis. You will only have a Cartesian plane if these two intersect. If it is a straight line graph, it is not a Cartesian plane.
First, a coordinate. A coordinate is a number. It labels a point on a line.Second, a coordinate axis is a line with coordinates.to label a point in a plane (a flat surface), we need more than one coordinate axis, and we place a second at right angles to the first.Those axes are called rectangular coordinate axes, because they are at right angles to one another. The coordinates on them are called rectangular coordinates. They are also called Cartesian coordinates.