Magnitude

Posts Tagged ‘magnitude’

Angles Between Vectors

Thursday, November 19th, 2009

Defining the Angles Between Vectors

Description

A detailed tutorial on how to define the angles between vectors. Step by step tutorial including several examples of angles between vectors for reference.

Overview

In general, it is easier to find the angle between 2D vectors, rather than 3D vectors. In order to define the angles between vectors, we need to use the dot product in conjunction with a few other functions. The angles between vectors can be expressed as angle = arccos(v1xv2), where v1xv2 is how the dot product is expressed.

Tags: 2D, 3D, absolute, algebra, angle, arccos, conjunction, cosine, define, degrees, dot, function, linear, magnitude, product, radians, value, vector
Posted in Algebra | No Comments »

Null Vector

Tuesday, October 27th, 2009

Definition of a Null Vector

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.

Tags: 0, algebra, arrow, coordinates, direction, Euclidean, length, magnitude, null, space, vector, zero
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Vector Equality

Tuesday, October 27th, 2009

Introduction to Vector Equality

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 ${\mathbf a} = a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3$ and ${\mathbf b} = b_1{\mathbf e}_1 + b_2{\mathbf e}_2 + b_3{\mathbf e}_3$ , they are equal if $a_1 = b_1,\quad a_2=b_2,\quad a_3=b_3.\,$ .

Tags: a, algebra, b, coordinates, direction, E, equal, equality, length, magnitude, vector
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Euclidean Vectors

Tuesday, October 27th, 2009

Overview of Euclidean 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.

Tags: algebra, arrow, cartesian, coordinate, direction, Euclidean, geometric, graph, initial, length, magnitude, point, system, terminal, vector
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Unit Vector

Friday, October 23rd, 2009

Definition of a Unit Vector

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:

$<span style="font-size: x-small;">\mathbf{\hat{a}} = \frac{\mathbf{a}}{\left\|\mathbf{a}\right\|} = \frac{a_1}{\left\|\mathbf{a}\right\|}\mathbf{e_1} + \frac{a_2}{\left\|\mathbf{a}\right\|}\mathbf{e_2} + \frac{a_3}{\left\|\mathbf{a}\right\|}\mathbf{e_3}</span>$

Tags: algebra, arbitrary, direction, formula, length, magnitude, normalizing, one, point, unit, vector
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Length of a Vector

Friday, October 23rd, 2009

How to Find the Length of a Vector

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:

$\<span style="font-size: x-small;">left\|\mathbf{a}\right\|=\sqrt{{a_1}^2+{a_2}^2+{a_3}^2}.</span>$

The Euclidean norm is a consequence of the Pythagorean theorem.

Tags: absolute value, algebra, consequence, Euclidean, length, magnitude, norm, pythagorean, theorem, vector
Posted in Algebra | No Comments »

Vector Subtraction

Friday, October 23rd, 2009

How to Solve Vectors Using Vector Subtraction

Description

A detailed tutorial on how to solve vectors using vector subtraction. Step by step tutorial including several examples of vector subtraction for reference.

Overview

Vector subtraction involves two vectors that do not have to be equal, and could have different magnitudes and directions. The vectors are referred to as a and b. The formula for vector subtraction is:

$\mathbf{a}-\mathbf{b}=(a_1-b_1)\mathbf{e_1}+(a_2-b_2)\mathbf{e_2}+(a_3-b_3)\mathbf{e_3}.$

In general, vector subtraction is defined geomtrically instead of algebraically, so it is not used quite as often as vector addition is.

Tags: addition, algebra, algebraically, direction, equal, formula, geometrically, Geometry, magnitude, subtraction, vector
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Vector Addition

Friday, October 23rd, 2009

How to Solve Vectors Using Vector Addition

Description

A detailed tutorial on how to solve vectors using vector addition. Step by step tutorial including several examples of vector addition for reference.

Overview

Vector addition involves two vectors that do not have to be equal, and could have different magnitudes and directions. The vectors are referred to as a and b. The formula for vector addition is:

$\mathbf{a}+\mathbf{b}=(a_1+b_1)\mathbf{e_1}+(a_2+b_2)\mathbf{e_2}+(a_3+b_3)\mathbf{e_3}.$

Vector addition is also occassionally referred to as the parallelogram rule, because on a picture diagram of vector addition the shape of a parallelogram is formed.

Tags: addition, algebra, direction, equal, formula, graph, magnitude, parallelogram, picture, rule, vector
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Tuesday, September 29th, 2009

Introduction to Magnitude

Description

A detailed tutorial of how to solve for magnitude. Step by step tutorial including several examples of how to solve for magnitude for reference.

Overview

The magnitude refers to size – in mathematical concepts, what is larger? What has a greater value or quantity? This is what you look for when arranging things in order of magnitude. Several different measurements are considered to be types of magnitude – examples are volume, area, and length. Things that can be ordered by magnitude are fractions, line segments, planes, solids, and angles. Magnitude is considered to be measured only in positive, not in negative – not to say that the absolute value is taken, just that negative numbers are not included.

Tags: angles, area, arithmetic, fractions, greater, length, line segments, magnitude, Math, measurement, planes, positive, solids, value, volume
Posted in Arithmetic | No Comments »