Posts Tagged ‘product’
Tuesday, October 27th, 2009
An Introduction to Relations
Description
A detailed tutorial on the introduction to relations. Step by step tutorial including several examples of the introduction to relations for reference.
Overview
A relation is defined as an ordered pair. However, that is not entirely accurate. A relation could either be an ordered pair or a set of ordered pairs. A relation can be used with either one or more normal sets, or one Cartesian product set. When used with a normal set, it is a set of ordered pairs. When used with a Cartesian product, it is the power set of that set.
Tags: cartesian, coordinates, discrete math, element, ordered pair, power, product, relation, set, subset, theory
Posted in Discrete Math | No Comments »
Tuesday, October 27th, 2009
Cartesian Products in Set Theory
Description
A detailed tutorial of Cartesian products in set theory. Step by step tutorial including several examples of Cartesian products in set theory for reference.
Overview
A Cartesian product is an operation that can be performed in set theory. It is named not for the multiplication that occurs, but for the way the resulting set is written: it is written in ordered pairs, just like Cartesian coordinates. Two sets are said to be multiplied, such as A and B. Whichever set is written first in the operation has its first coordinate written with the second coordinate of the second set. This continues until all coordinates have been used at least once.
Tags: cartesian, coordinates, discrete math, element, multiplication, operation, ordered pair, product, set, subset, theory
Posted in Discrete Math | No Comments »
Tuesday, October 27th, 2009
Introduction to Hilbert Space
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.
Tags: abstract, algebra, angle, Calculus, complete, Hilbert, inner, length, limit, product, space, structure, vector
Posted in Algebra | No Comments »
Tuesday, October 27th, 2009
Definition of a Scalar Triple Product
Description
A detailed tutorial on scalar triple products. Step by step tutorial including several examples of scalar triple products for reference.
Overview
A scalar triple product is a way of applying other multiplication operators to three vectors. Quite often, the scalar triple product is denoted as (a, b, c). It can also be defined as (a b c) = a(b x c). The scalar triple product has three main properties. The first one is that the absolute value of the scalar triple product is the volume of the three dimensional figure that is formed by the three vectors. The second one is the scalar triple product is only zero if the three vectors are linearly independent. The three vectors must lie in the same plane for this to be true. The third one is that the scalar triple product is only positive if all three of the vectors are considered right-handed.
A simple way to write the scalar triple product is to line up the coordinates of the vectors in this form: 
This is the same as saying 
Tags: absolute, algebra, box, coordinates, figure, independent, linear, mixed, multiplication, operator, parallelpiped, positive, product, properties, right-handed, scalar, three-dimensional, triple, value, zero
Posted in Algebra | No Comments »
Tuesday, October 27th, 2009
The Cross Product of Vectors
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 
. 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.
Tags: a, algebra, b, cross, dot, n!, outer, perpendicular, product, right-handed, rule, scalar, system, unit, vector
Posted in Algebra | No Comments »
Friday, October 23rd, 2009
Overview of the Dot Product
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>")
Tags: algebra, algebraically, angle, common, component, cosine, direction, dot, geometrically, initial, inner, length, mulitplied, point, product, scalar, starting, vector
Posted in Algebra | No Comments »
Friday, September 25th, 2009
How to Simplify Factorials
Description
A detailed tutorial on how to simplify factorials. Step by step tutorial including several examples of how to simplify factorials for reference.
Overview
A factorial is an interesting mathematical function. It is expressed as a number with an exclamation point after it – for example, 5! would be “five factorial”. What a factorial really is, is an expression of multiplication. In n!, all numbers from 1 to n, including n, are multiplied. For example: 7! = 1 * 2 * 3 * 4 * 5 * 6 * 7. The notation of a factorial was thought up by Christian Kramp in 1808.
Tags: algebra, Christian Kramp, factorial, Math, multiplication, multiply, n!, product, simplify
Posted in Algebra | No Comments »
Tuesday, September 15th, 2009
How to Find a Percentage of Any Number
Description
A detailed tutorial on the finding of percentages of any number. Step by step tutorial including several examples of how to find percents for reference.
Overview
The knowledge of how to find percents isn’t just something you’ll find in the classroom. You’ll find it all over for the rest of your life – at home, at work, even at the grocery store. Finding percents is a very important skill. It can be very confusing to be faced with some percent of a number that isn’t 100. Thankfully, it is very easy to solve, with the help of cross-multiplication. Cross-multiplication is setting up two fractions and multiplying them in an x formation. One of your fractions will be the percent – a number over 100 – and the other fraction will be an unknown variable over the number you need to find the percent of. After using cross-multiplication, just solve for the unknown variable. The video provided shows a second method to solve percents with.
Tags: algebra, cross multiplication, cross multiply, Math, multiplication, multiply, number, percentage, percents, product
Posted in Algebra | No Comments »
Tuesday, September 15th, 2009
How to Multiply Decimals
Description
A detailed tutorial on how to multiply decimals. Step by step tutorial including several examples of multiplying decimals for reference. It is a requirement to know how to multiply decimals for all math classes.
Overview
Decimals are really no different from regular numbers when you perform operations on them, but sometimes the numbers in the decimal places can be a little tricky to figure out. The operation we will be talking about is multiplication. Normally, when performing an operation on decimals, you match up the decimal points. However, in multiplication you pretend that the decimal points don’t exist. You multiply as you normally would. However, you do need a decimal point in your final answer. You you need to perform a second operation. Count how many decimal places are in your first decimal, and then count how many there are in your second decimal. Add them together. When you get your final answer, count that many numbers (starting from the right) and then put down your decimal point that many places over.
Tags: arithmetic, decimal points, decimals, Math, multiplication, multiply, operations, point, product
Posted in Arithmetic | No Comments »
Tuesday, September 15th, 2009
An In-Depth Look at the Closure Property
Description
A detailed tutorial on how to use the closure property. Step by step tutorial including several examples of how to use the closure property for reference.
Overview
The closure property states that if a and b are both real numbers, then a + b is a unique real number, and a * b is also a unique real number. Basically what the closure property is saying is that if you add or multiply two real numbers, your only possible answer is a real number. The closure property is also saying that the sum or product of two real numbers is unique, meaning there is only one number that it could be.
Tags: add, addition, arithmetic, closure, closure property, Math, multiplication, multiply, product, property, real numbers, sum, unique
Posted in Arithmetic | No Comments »
