5 Ways To Normalize A Vector - WikiHow

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Terms of Use wikiHow is where trusted research and expert knowledge come together. Learn why people trust wikiHow How to Normalize a Vector PDF download Download Article Explore this Article parts 1 Define Terms 2 Analyze the Objective 3 Derive a Solution for the Unit Vector 4 Normalize a Vector in 2 Dimensional Space 5 Normalize a Vector in n Dimensional Space + Show 2 more... - Show less... Other Sections VideoWATCH NOW Questions & Answers Related Articles References Article Summary Reviewed by Joseph Meyer

Last Updated: February 19, 2026 Fact Checked

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This article was reviewed by Joseph Meyer. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 267,533 times.

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

Part 1 Part 1 of 5:

Define Terms

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  1. Step 1 Define a unit vector. 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.[1] It can be mathematically proven that there is one and only one unit vector for each given vector A.
  2. Step 2 Define the Normalization of a vector. 2 Define the Normalization of a vector. This is the process of identifying the unit vector for a given vector A.[2] Advertisement
  3. Step 3 Define a bound vector. 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.[3]
  4. Step 4 Describe vector notation. 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.[4]
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Part 2 Part 2 of 5:

Analyze the Objective

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  1. Step 1 Establish the known values. 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.[5]
  2. Step 2 Determine the unknown value. 2 Determine the unknown value. The only variable we need to calculate is the terminal point of the unit vector.
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Part 3 Part 3 of 5:

Derive a Solution for the Unit Vector

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  1. Step 1 Find the terminal point for the unit vector of vector A = (x, y). 1 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.[6] 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))
Part 4 Part 4 of 5:

Normalize a Vector in 2 Dimensional Space

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  1. Step 1 Let vector A be a vector with its initial point at the origin and terminal point at (2,3), such that A = (2,3). 1 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))).
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Part 5 Part 5 of 5:

Normalize a Vector in n Dimensional Space

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  1. 1 Generalize the equation for vector normalization in space of any dimension.[7] 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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References

  1. ↑ https://mathworld.wolfram.com/UnitVector.html
  2. ↑ https://www.khanacademy.org/computing/computer-programming/programming-natural-simulations/programming-vectors/a/vector-magnitude-normalization
  3. ↑ https://webspace.maths.qmul.ac.uk/j.n.bray/MTH4103/GeomINotes01.pdf
  4. ↑ https://engcourses-uofa.ca/books/statics/vectors-and-their-operations/cartesian-vector-notation/
  5. ↑ https://mathworld.wolfram.com/NormalizedVector.html
  6. ↑ https://www.khanacademy.org/computing/computer-programming/programming-natural-simulations/programming-vectors/a/vector-magnitude-normalization
  7. ↑ https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematics_for_Game_Developers_(Burzynski)/03%3A_Vectors_in_Three_Dimensions/3.04%3A_The_Unit_Vector_in_3-Dimensions_and_Vectors_in_Standard_Position

About This Article

Joseph Meyer Reviewed by: Joseph Meyer Math Teacher This article was reviewed by Joseph Meyer. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 267,533 times. 12 votes - 45% Co-authors: 12 Updated: February 19, 2026 Views: 267,533 Categories: Geometry Article SummaryX

A vector is an object that has both magnitude (sometimes called size or length) and direction. Vectors are usually represented by drawing an arrow, where the direction of the arrow represents the direction of the vector, and the length of the arrow represents its magnitude. Normalizing a vector involves converting it to a “unit vector” with a standard magnitude, usually 1, while preserving the vector’s original direction. To do this, start by determining the start and end points of your vector. For instance, the vector may start at (0,0) on the x-y axis, and end at (3,4). This vector moves up from left to right. From there, you can determine that your unit vector will have the same starting point and direction as the original vector. You also know that the length of your unit vector is 1. Now you’ll need to calculate the end point, or terminal point, of your unit vector. First, calculate the length of the original vector using the Pythagorean theorem, a^2 + b^2 = c^2. Think of the vector as a right triangle, where sides A and B equal the values of the end coordinates in the x and y axes, and the hypotenuse is the length of the vector. In this case, we know that 32 + 42 = 25. Take the square root of 25 to get 5, the length of the vector. Finally, divide the x and y coordinates by the length of the vector to get the endpoint coordinates of your normalized vector. Now you know that the start point of your unit vector is (0,0), its end point is (3/5, 4/5), its magnitude is 1, and it moves up as you go from left to right along the x axis. To learn how to normalize a vector in 2-dimensional or n-dimensional space, keep reading! Did this summary help you?YesNo

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