Website owner: James Miller
Vectors in n-dimensional space. Orthogonal vectors. Pythagorean theorem. Orthogonal and orthonormal bases.
Length of a vector in n-dimensional space. If x is a vector in n-space, its length is given by
i.e. the square root of the dot product xx.
Distance between points in n-space. If x and y represent two points in n-space the distance between them is given by
(i.e. the length of the vector x - y )
Angle between two vectors in n-space. If x and y are two vectors in n-space the angle θ between them is given by
This formula is a direct generalization of the formula xy = ||x|| ||y|| cos θ for three dimensional space. By the Schwarz inequality (theorem),
|xy| ||x|| ||y||
for any two n-vectors x and y. Thus the fraction
is always in the range -1 to +1.
Orthogonal vectors in n-space. Two vectors x and y in n-space are said to be orthogonal if and only if
(i.e. they are orthogonal if cos θ = 0, and thus θ = 90o.)
Pythagorean theorem in n-space. Let x1, x2, ...... , xk be k pairwise orthogonal vectors in n-space. Then
( x1 + x2 + ...... + xk ) · ( x1 + x2 + ...... + xk ) = x1 · x1 + x2 · x2 + ...... + xk · xk
i.e. the square of the diagonal of the k-dimensional rectangular parallelepiped formed by x1, x2, ...... , xk is equal to the sum of the squares of the edges. See Fig. 1 for case k = 3.
Basis for an n-dimensional space. A basis for an n-dimensional space is any set of linearly independent vectors that span the space. Every vector in the space the can be expressed as a unique linear combination of the vectors in this basis.
Orthogonal basis for an n-dimensional space. An orthogonal basis for an n-dimensional space is any set of n pairwise orthogonal vectors in the space. Every vector in the space the can be expressed as a unique linear combination of the vectors in this basis.
Orthonormal basis for an n-dimensional space. An orthonormal basis for an n-dimensional space is any set of n pairwise orthogonal unit vectors in the space. If x1, x2, ...... , xn are a set of n pairwise orthogonal vectors in the space, than the set
constitutes an orthonormal basis for the space. Every vector in the space the can be expressed as a unique linear combination of the vectors in this basis.
Vectors referred to an orthonormal basis. Let the vectors x1, x2, ...... , xn represent an orthonormal basis for an n-dimensional space. Then any vector x in the space can be represented as
x = c1x1 + c2x2 + .......... + cnxn
where c1, c2, ...... , cn are constants. The coefficients c1, c2, ...... , cn represent the projections of the vector x on the basis vectors x1, x2, ...... , xn (they can be viewed as the coordinates of point x in the orthonormal system).
The Way of Truth and Life
God's message to the world
Jesus Christ and His Teachings
Words of Wisdom
Way of enlightenment, wisdom, and understanding
Way of true Christianity
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Who will go to heaven?
The superior person
On faith and works
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
The teaching is:
On modern intellectualism
On Self-sufficient Country Living, Homesteading
Principles for Living Life
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
America has lost her way
The really big sins
Theory on the Formation of Character
You are what you eat
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
These things go together
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
The True Christian
What is true Christianity?
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Walking a solitary road
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
On responding to wrongs
Real Christian Faith
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
Sizing up people
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life
We are our habits
What creates moral character?