SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

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 x•x.

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 x•y = ||x|| ||y|| cos θ for three dimensional space. By the Schwarz inequality (theorem),

|x•y| ||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 = 90^{o}.)

Pythagorean theorem in n-space. Let x_{1}, x_{2}, ...... , x_{k} be k
pairwise orthogonal vectors in n-space. Then

( x_{1 }+ x_{2} + ...... + x_{k} ) · ( x_{1 }+
x_{2} + ...... + x_{k} ) = x_{1} ·_{ }x_{1} + x_{2} · x_{2} +
...... + x_{k} · x_{k}

i.e. the square of the diagonal of the
k-dimensional rectangular
parallelepiped formed by x_{1}, x_{2}, ......
, x_{k} 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 x_{1}, x_{2}, ...... , x_{n
}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 x_{1}, x_{2}, ...... , x_{n } represent an
orthonormal basis for an n-dimensional space. Then any vector x in the space can be represented
as

x = c_{1}x_{1} + c_{2}x_{2} + .......... + c_{n}x_{n}

where c_{1}, c_{2}, ...... , c_{n} are constants. The coefficients c_{1}, c_{2}, ...... , c_{n} represent the projections of
the vector x on the basis vectors x_{1}, x_{2}, ...... , x_{n } (they can be viewed as the coordinates of point x
in the orthonormal system).

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

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

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

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

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

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

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

[ Home ] [ Up ] [ Info ] [ Mail ]