Unit Vector
Keep the direction, set the length to 1 · 단위벡터
1. What Is a Unit Vector?
A unit vector is a vector with magnitude exactly 1. It carries direction only — all information about length has been removed. A unit vector is usually written with a hat, â (“a-hat”).
Turning any nonzero vector into a unit vector is called normalizing. It is how you answer “which way does this point?” without caring how long the arrow is — essential for directions, normals, and headings.
단위벡터는 크기가 정확히 1인 벡터로, 길이 정보는 버리고 방향만 담습니다(보통 â로 표기). 임의의 0이 아닌 벡터를 단위벡터로 만드는 것을 정규화라 하며, "어느 방향인가"를 길이와 무관하게 다룰 때 씁니다.
2. The Normalization Formula & a Worked Example
Divide the vector by its own magnitude:
â = a / |a| = (a₁ / |a|, a₂ / |a|, …, aₙ / |a|)
Worked example. For a = (2, 3, 6), first the magnitude: |a| = √(4 + 9 + 36) = √49 = 7. Then divide each component by 7:
â = (2/7, 3/7, 6/7)
Check: |â| = √((2/7)² + (3/7)² + (6/7)²) = √(49/49) = 1 — exactly as required.
벡터를 자신의 크기로 나눕니다: â = a / |a|. 예: a = (2, 3, 6), |a| = 7 → â = (2/7, 3/7, 6/7). 검산하면 |â| = √(49/49) = 1 로 길이가 정확히 1입니다.
3. The Geometric Picture & Standard Basis
Normalizing slides the tip in or out along the same line until the arrow has length 1 — direction unchanged, size standardized.
The most common unit vectors are the standard basis vectors along each axis: i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1). Any vector can be rebuilt from its length and its unit direction: a = |a| · â.
정규화는 같은 직선을 따라 끝점을 옮겨 길이를 1로 맞춥니다(방향 불변). 대표적 단위벡터는 표준기저 i = (1,0,0), j = (0,1,0), k = (0,0,1)이며, 임의의 벡터는 a = |a| · â 로 복원됩니다.
4. Key Properties
- Length 1. |â| = 1 by construction.
- Same direction. â points exactly the same way as a (it is a positive scalar multiple).
- Reconstruct the original. a = |a| · â separates size from direction.
- The zero vector cannot be normalized. It has no direction, and dividing by |0| = 0 is undefined.
단위벡터는 |â| = 1, a와 같은 방향(양의 스칼라배), a = |a| · â 로 원래 벡터 복원이 가능합니다. 단, 영벡터는 방향이 없고 0으로 나눌 수 없어 정규화할 수 없습니다.
5. Frequently Asked Questions
How do you find a unit vector? Divide the vector by its magnitude: â = a / |a|. The result keeps the direction but has length 1.
Can you normalize the zero vector? No. It has magnitude 0 and no defined direction, so a / |a| is undefined.
What is a unit vector used for? Representing pure direction — surface normals, camera headings, and the direction part of any vector via a = |a| · â.
벡터를 크기로 나누면(â = a / |a|) 방향은 같고 길이가 1인 단위벡터가 됩니다. 영벡터는 크기가 0이라 정규화할 수 없습니다. 단위벡터는 순수한 방향을 표현하는 데 쓰입니다.
Ready to practice? Drill unit vectors and the rest of linear algebra on C:Vector, or review the full vector reference and the related magnitude, scalar multiplication, and projection.