A vector is a geometric object that has a direction and a magnitude. It is represented as an oriented segment with a starting point and an arrow on the opposite end; the length of the segment is proportional to the magnitude and the direction of the arrow indicates the direction. Vector normalization is a fairly common exercise in mathematics and has several practical applications in computer graphics.
Steps
Method 1 of 5: Define the Terms
Step 1. Define the unit vector or vector unit
The vector of vector A is precisely a vector that has the same direction and direction as A, but length equal to 1 unit; it can be proved mathematically that for each vector A there is only one unit vector.
Step 2. Define the normalization of a vector
It is a question of identifying the unit vector for that A given.
Step 3. Define the applied vector
It is a vector whose starting point coincides with the origin of the coordinate system within a Cartesian space; this origin is defined with the pair of coordinates (0, 0) in a two-dimensional system. This way, you can identify the vector by referring only to the end point.
Step 4. Describe vector notation
Limiting yourself to the vectors applied, you can indicate the vector as A = (x, y), where the pair of coordinates (x, y) defines the end point of the vector itself.
Method 2 of 5: Analyze the Goal
Step 1. Establish known values
From the definition of versor you can deduce that the starting point and the direction coincide with those of the given vector A; moreover, you know for sure that the length of the vector unit is equal to 1.
Step 2. Determine the unknown value
The only variable you need to calculate is the end point of the vector.
Method 3 of 5: Derive the Solution for the Unit Vector
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Find the end point of the vector unit A = (x, y). Thanks to the proportionality between similar triangles, you know that every vector that has the same direction of A has as its terminal the point with coordinates (x / c, y / c) for each value of "c"; moreover, you know that the length of the vector unit is equal to 1. Consequently, using 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); it follows that the vector u of the vector A = (x, y) is defined as u = (x / (x ^ 2 + y ^ 2) ^ (1/2), y / (x ^ 2 + y ^ 2) ^ (1/2))
Normalize to Vector Step 6
Method 4 of 5: Normalize a Vector in a Two-dimensional Space
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Consider the vector A whose starting point coincides with the origin and the final one with the coordinates (2, 3), consequently 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))). Hence, A = (2, 3) normalizes to u = (2 / (13 ^ (1/2)), 3 / (13 ^ (1/2))).
Normalize to Vector Step 6
