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Home Vectors Hyperreal Vectors Theorem 2: Real Length and Real Direction
 
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Theorem 2: Real Length and Real Direction

Two new concepts which arise in the study of hyperreal vectors are vectors with real length and vectors with real direction. We say that A has real length if |A| is a real number. We say that A has real direction if the unit vector of A is real, or equivalently, the direction cosines of A are real.

There are four types of hyperreal vectors:

  1. Vectors with real length and real direction.
  2. Vectors with real length but nonreal direction.
  3. Vectors with nonreal length but real direction.
  4. Vectors with nonreal length and nonreal direction.

THEOREM 2

A vector is real if and only if it has both real length and real direction.

PROOF

A has real length and direction if and only if |A| and U = A/|A| are both real if and only if A = |A|U is real.

Example 2

Two hyperreal vectors A and B with unit vectors U and V are said to be almost parallel if either U ≈ V or U ≈ -V.

Example 3

Home Vectors Hyperreal Vectors Theorem 2: Real Length and Real Direction

Last Update: 2006-11-05