Description
A detailed tutorial on the definition of an initial point. Step by step tutorial including several examples of initial points for reference.
Overview
An initial point is just a way of saying the starting point. The initial point of a line or a figure is the point where it begin. The term initial point is used often when talking about vectors – they start at the initial point. The initial point is referred as the tail of the vector.
Description
A detailed tutorial on on the application of Hilbert space. Step by step tutorial including several examples of Hilbert space for reference.
Overview
A Hilbert space is commonly used in vector algebra and calculus to generalize the notion of Euclidean space. It is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Hilbert spaces are also complete, which is a property that allows enough limits in the space for calculus to be used accurately.
Description
A detailed tutorial on finding the component of a vector. Step by step tutorial including several examples of how to find the component of a vector for reference.
Overview
There are three different kinds of components that can be found in vectors: axial components, radial components, and tangential components. Just like the vectors themselves, different types of components are found in different coordinate systems. Axial components are found in the Cartesian coordinate system, while radial and tangential components are found in the polar coordinate system. A component is exactly the same as it’s dictionary definition: it is just a small part that makes up a whole, so in this case they are small parts of a vector. The vector itself is also a component.
Description
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.
Overview
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 
Description
A detailed tutorial on the definition of a null vector. Step by step tutorial including several examples of null vectors for reference.
Overview
A null vector is a vector that has no direction. It is placed at the coordinates (0, 0, 0) in Euclidean space. Another name for a null vector is a zero vector. Although the null vector is the only vector that has no direction, we cannot say that the null vector is unique because more than one vector has the possibility of being null.
Description
A detailed tutorial on how to determine if two vectors are equal. Step by step tutorial including several examples of vector equality for reference.
Overview
Vectors are said to be equal if they have the same magnitude and direction. They must also have the same coordinates. Using this logic, it is possible to determine if you have two vectors 
Description
A detailed tutorial on Euclidean vectors. Step by step tutorial including several examples and visual examples of Euclidean vectors for reference.
Overview
A vector is a geometric object that has both a magnitude (also known as the length) and a direction. They are usually drawn as arrows that have a similar starting point and connect two points together. The difference between different kinds of vectors is what coordinate system is used to describe them. Euclidean vectors are vectors that are described by the Cartesian coordinate system.
Description
A detailed tutorial on the four-vector. Step by step tutorial including several examples of the four-vector and how to solve it for reference.
Overview
In linear algebra, a four-vector is defined as a vector in four-dimensional real vector space. The difference between a vector and a four-vector is that a four-vector can be transformed by Lorentz transformations. The concept of four-vectors branches out throughout vector mathematics, and to special relativity and general relativity. The concepts are a little different in each, but not enough to make it confusing if you are learning about them from a different standpoint. Four-vectors are often used in combination with matrices.
Description
A detailed tutorial on the unit vector. Step by step tutorial including several examples of the unit vector and how to solve it for reference.
Overview
In linear algebra, a unit vector is a vector that only has a length or magnitude of one. They are often used to indicate direction. There is a process used to create a unit vector, called normalizing a vector. When doing this, you must divide a vector of arbitrary length by its length. To normalize a vector with three points, you would use this formula:
Description
A detailed tutorial of the dot product. Step by step tutorial including several examples of the dot product of a vector for reference.
Overview
The dot product of two vectors always ends up being a scalar. In mathematical terms, this is
![<span style="font-size: x-small;">\mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta[</span>/latex]. In this case, theta is the measure of the angle between a and b. The definition of a dot product given geometrically is that a and b have a common starting point and that the length of a is multiplied by the component in b that points in the same direction as a. Algebraically, it can be said that [latex]<span style="font-size: x-small;">\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3.</span>](http://s.wordpress.com/latex.php?latex=%3Cspan%20style%3D%22font-size%3A%20x-small%3B%22%3E%5Cmathbf%7Ba%7D%5Ccdot%5Cmathbf%7Bb%7D%3D%5Cleft%5C%7C%5Cmathbf%7Ba%7D%5Cright%5C%7C%5Cleft%5C%7C%5Cmathbf%7Bb%7D%5Cright%5C%7C%5Ccos%5Ctheta%5B%3C%2Fspan%3E%2Flatex%5D.%20In%20this%20case%2C%20theta%20is%20the%20measure%20of%20the%20angle%20between%20a%20and%20b.%20The%20definition%20of%20a%20dot%20product%20given%20geometrically%20is%20that%20a%20and%20b%20have%20a%20common%20starting%20point%20and%20that%20the%20length%20of%20a%20is%20multiplied%20by%20the%20component%20in%20b%20that%20points%20in%20the%20same%20direction%20as%20a.%20Algebraically%2C%20it%20can%20be%20said%20that%20%5Blatex%5D%3Cspan%20style%3D%22font-size%3A%20x-small%3B%22%3E%5Cmathbf%7Ba%7D%20%5Ccdot%20%5Cmathbf%7Bb%7D%20%3D%20a_1%20b_1%20%2B%20a_2%20b_2%20%2B%20a_3%20b_3.%3C%2Fspan%3E&bg=ffffff&fg=000000&s=0 "<span style=\"font-size: x-small;\">\mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta[</span>/latex]. In this case, theta is the measure of the angle between a and b. The definition of a dot product given geometrically is that a and b have a common starting point and that the length of a is multiplied by the component in b that points in the same direction as a. Algebraically, it can be said that [latex]<span style=\"font-size: x-small;\">\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3.</span>")
