SolitaryRoad.com

Website owner: James Miller

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

LINES AND PLANES IN SPACE

Equation of a line in space. Let be the position vector of a point P in space. Then the vector equation of a line passing through point P in a direction given by vector is

where t is a parameter that ranges over the real numbers. If

then 1) is equivalent to the following parametric representation of a straight line:

As the parameter t varies over the real numbers the vector generates the line.

Equation of a plane in space. Let be the position vector of a point P in space. Then the vector equation of a plane passing through point P and parallel to two noncollinear vectors and is

where parameters u and v range independently over the real numbers. If

then 3) is equivalent to the following parametric representation of a plane:

As the parameters u and v vary over the real numbers the vector generates the plane.

Normal and one point form of the equation of a plane. Let be the position vector of a point P in space. Then the vector equation of a plane passing through point P and perpendicular to some vector (i.e. a normal to the plane) is

If

5) is equivalent to

6) (x_{1} - a_{1})n_{1} + (x_{2} - a_{2})n_{2} + (x_{3} - a_{3})n_{3} = 0

Three point form of the equation of a plane. Let
be the position vectors of
three noncollinear points P_{1}, P_{2} and P_{3} in space. Then the vector equation of the plane passing
through points P_{1}, P_{2} and P_{3} is

Proof. The vectors and represent two linearly independent vectors in the plane and so represents a vector normal to the plane. Equation 7) then follows from equation 5) above.

If

7) is equivalent to

Reference.

Lipschutz. Differential Geometry. Chap. 2

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 ]