Normalize a Vector

A vector is a geometric object that has direction and magnitude. It may be represented as a line segment with an initial point (starting point) on one end and an arrow on the other end, such that the length of the line segment is the magnitude of the vector and the arrow indicates the direction of the vector. Vector normalization is a common exercise in mathematics and it also has practical applications in computer graphics.

Steps

Define Terms

  1. Define a unit vector. The unit vector of a vector A is the vector with the same initial point and direction as A, but with a length of 1 unit. It can be mathematically proven that there is one and only one unit vector for each given vector A.
  2. Define the Normalization of a vector. This is the process of identifying the unit vector for a given vector A.
  3. Define a bound vector. A bound vector in Cartesian space has its initial point at the origin of the coordinate system, expressed as (0,0) in two dimensions. This allows you to identify a vector solely in terms of its terminal point.
  4. Describe vector notation. By restricting ourselves to bound vectors, A = (x, y) where the coordinate pair (x,y) indicates the location of the terminal point for vector A.

Analyze the Objective

  1. Establish the known values. From the definition of the unit vector, we know that the initial point and direction of the unit vector is the same as the given vector A. Furthermore, we know the length of the unit vector is 1.
  2. Determine the unknown value. The only variable we need to calculate is the terminal point of the unit vector.

Derive a Solution for the Unit Vector

  • Find the terminal point for the unit vector of vector A = (x, y). From the proportionality of similar triangles, you know that any vector that has the same direction as vector A will have a terminal point (x/c, y/c) for some c. Furthermore, you know the length of the unit vector is 1. Therefore, by the Pythagorean Theorem, [x^2/c^2 + y^2/c^2]^(1/2) = 1 -> [(x^2 + y^2)/c^2]^(1/2) -> (x^2 + y^2)^(1/2)/c = 1 -> c = (x^2 + y^2)^(1/2). Therefore, the unit vector u for the vector A = (x, y) is given as u = (x/(x^2 + y^2)^(1/2), y/(x^2 + y^2)^(1/2))

Normalize a Vector in 2 Dimensional Space

  • Let vector A be a vector with its initial point at the origin and terminal point at (2,3), such that A = (2,3). Calculate the unit vector u = (x/(x^2 + y^2)^(1/2), y/(x^2 + y^2)^(1/2)) = (2/(2^2 + 3^2)^(1/2), 3/(2^2 + 3^2)^(1/2)) = (2/(13^(1/2)), 3/(13^(1/2))). Therefore, A = (2,3) normalizes to u = (2/(13^(1/2)), 3/(13^(1/2))).

Normalize a Vector in n Dimensional Space

  • Generalize the equation for vector normalization in space of any dimension. A vector A (a, b, c, …), u = (a/z, b/z, c/z, …) where z = (a^2 + b^2 + c^2 …)^(1/2).



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