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A detailed tutorial of the x axis and the  y axis. Step by step tutorial including several examples of the x axis and the y axis for reference.

The the Cartesian coordinate system, there is an x axis and a y axis. The x axis runs horizontally across the system and all first terms in ordered pairs are x coordinates, from the x axis. The y axis runs vertically across the system and all second terms in ordered pairs are y coordinates, from the y axis. The x and y axis work together to use a pattern of right angles and perpendicular lines in order to find ordered pairs and coordinates of x and y on the graph.

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A detailed tutorial on how to find the altitude of a triangle. Step by step tutorial including several examples of how to find the altitude of a triangle for reference.

The altitude is just a way of saying the height of something. Typically, the term altitude is only used to refer to triangles. In triangles, the altitude is a little different from the height. Unlike the height, the altitude can be taken from three points of the triangle – it can be taken through the center of any of the three vertexes of the triangle. The altitude goes from the vertex to the line across from it, forming a right angle with that line. All three altitudes should intersect at a common point in the center of the triangle, known as the orthocenter.

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A detailed tutorial on the definition of a bisector. Step by step tutorial including several examples of bisectors for reference.

A bisector is any line that evenly divides a symmetrical shape or object. The only difference between the bisector and the test for symmetry is that when testing for symmetry, the line is not really there. A bisector is really there. The most common kind of bisector is an angle bisector. In order to remember bisectors, think of them as perpendicular lines that cross right in the middle.

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A detailed tutorial on orthogonal vectors. Step by step tutorial including several examples of orthogonal vectors for reference.

Orthogonal vectors are vectors that are perpendicular. You can determine if vectors are perpendicular by finding the dot product. If the dot product is equal to zero, then the vectors are perpendicular. In certain dimensions, it is possible for three vectors to be perpendicular to each other. In this case, all three of those vectors are considered to be orthogonal. However, in general, orthogonal vectors is a term used to describe a pair of vectors.

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A detailed tutorial on orthogonal complements. Step by step tutorial including several examples of orthogonal complements for reference.

The orthogonal complement of a subspace of an inner product space is the set of all vectors in the inner product space that are orthogonal to every vector in the subspace. This can be expressed mathematically in the formula , where W is the subspace and V is the inner product space. The orthogonal complement is sometimes also called the perpendicular complement, shortened to the informal form perp.

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A detailed tutorial on identifying coprime numbers. Step by step tutorial including several examples of how to identify coprime numbers for reference.

Two numbers are considered to be coprime, or relatively prime, if they have no common positive factor other than 1, or if their greatest common divisor is 1. Sometimes the notation for perpendicular is used to say that a number A is coprime to another number B. The term coprime was invented because the numbers are prime together, but are not prime themselves. A prime number can be coprime with any number.

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A detailed tutorial on the cross product of two vectors. Step by step tutorial including several examples of how to find the cross product for reference.

A cross product is very similar to a dot product. However, the result of a cross product is a vector, and the result of a dot product is a scalar. In mathematical terms, the cross product can be defined as . Theta represents the meausre of the angle between a and b, and n is a unit vector perpendicular to both a and b. The vector this forms is a right-handed system.

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A detailed tutorial on identifying points, lines, and planes. Step by step tutorial including several examples of how to indentify different types of points, lines, and planes for reference.

Points, lines and planes might not seem very similar, but they are all connected. Points can be found in or on a line or plane, and lines form the planes. Points are simply locations, and are represented by small dots. There are a few different kinds of points. Collinear points are points that lie on the same line, while noncollinear points are points that don’t lie on the same line. Coplaner points lie in the same plane, while noncoplaner points do not.  Lines have no thickness to them and extend infinitely in both directions. There are several different types of lines. There are skew lines, which run next to each other (although not parallel) but never touch each other. There are parallel lines and perpendicular lines, and there are intersecting lines. Planes have no thickness and are perfectly flat figures represented by a rectangle shape. Lines and planes can run in many different directions with each other, but have no special names.