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Aspect Ratio

Christine — Sat, 14 Nov 2009 00:49:11 +0000

Introduction to Aspect Ratio

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Description

A detailed tutorial on what aspect ratio is. Step by step tutorial including several examples of how to find the aspect ratio for reference.

Overview

The aspect ratio can only be used when referring to a shape, typically a square type of shape, such as a square, rhombus, rectangle, or parallelogram. The aspect ratio is used very often for describing measurements. It is the ratio of the longer dimension to the shorter dimension – that is, the length to the width. In a 3D shape, the depth – which is the second measurement of width – is added to the end of this measurement.

Hilbert Space

Christine — Tue, 27 Oct 2009 21:17:38 +0000

Introduction to Hilbert Space

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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.

Null Vector

Christine — Tue, 27 Oct 2009 18:39:55 +0000

Definition of a Null Vector

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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.

Vector Equality

Christine — Tue, 27 Oct 2009 18:01:49 +0000

Introduction to Vector Equality

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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 and , they are equal if .

Euclidean Vectors

Christine — Tue, 27 Oct 2009 17:05:50 +0000

Overview of Euclidean Vectors

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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.

Unit Vector

Christine — Fri, 23 Oct 2009 23:45:01 +0000

Definition of a Unit Vector

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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:

\mathbf{\hat{a}} = \frac{\mathbf{a}}{\left\|\mathbf{a}\right\|} = \frac{a_1}{\left\|\mathbf{a}\right\|}\mathbf{e_1} + \frac{a_2}{\left\|\mathbf{a}\right\|}\mathbf{e_2} + \frac{a_3}{\left\|\mathbf{a}\right\|}\mathbf{e_3}' title='\mathbf{\hat{a}} = \frac{\mathbf{a}}{\left\|\mathbf{a}\right\|} = \frac{a_1}{\left\|\mathbf{a}\right\|}\mathbf{e_1} + \frac{a_2}{\left\|\mathbf{a}\right\|}\mathbf{e_2} + \frac{a_3}{\left\|\mathbf{a}\right\|}\mathbf{e_3}' class='latex' />

Dot Product

Christine — Fri, 23 Oct 2009 23:38:33 +0000

Overview of the Dot Product

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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 \mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta[/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]\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3.' title='\mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta[/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]\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3.' class='latex' />

Length of a Vector

Christine — Fri, 23 Oct 2009 22:57:46 +0000

How to Find the Length of a Vector

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Description

A detailed tutorial on finding the length of a vector. Step by step tutorial including several examples of how to find the length of a vector for reference.

Overview

The length of a vector is also known as the magnitude of a vector. This can be compared to the absolute value of a real number. In order to find the length of a vector, you need to use the Euclidean norm:

left\|\mathbf{a}\right\|=\sqrt{{a_1}^2+{a_2}^2+{a_3}^2}.' title='\left\|\mathbf{a}\right\|=\sqrt{{a_1}^2+{a_2}^2+{a_3}^2}.' class='latex' />

The Euclidean norm is a consequence of the Pythagorean theorem.

Law of Tangents

Christine — Thu, 08 Oct 2009 22:41:00 +0000

An Introduction to the Law of Tangents

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Description

A detailed tutorial on the Law of Tangents. Step by step tutorial including several examples of the Law of Tangents for reference.

Overview

The Law of Tangents refers to the lengths of the three sides of a triangle and the tangents of the angles. This can be used with respect to any triangle, not just right triangles. While the Law of Tangents is not as well known as the Law of Sines or the Law of Cosines, it is useful. The Law of Tangents can be used whenever either two sides and an angle, or two angles and a side, are known on any given triangle. The proof of this law starts with the Law of Sines. The Law of Tangents is as follows:

Parallelogram Law

Christine — Thu, 01 Oct 2009 22:38:33 +0000

Introduction to the Parallelogram Law

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Description

A detailed tutorial of the parallelogram law. Step by step tutorial including several examples of the parallelogram law for reference.

Overview

The parallelogram law shows up in many forms, but the simplest form states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. Assuming that a rectangle has four corners A, B, C, and D, this can be expressed as:

Typically, the two diagonals of a parallelogram are not equal in length. If they are, then the equation simplifies to the Pythagorean theorem. A more complicated version of the parallelogram law is often found when calculating vectors.