Magnitude

1. What Is the Magnitude of a Vector?

The magnitude of a vector — also called its length or norm, written |a| or ‖a‖ — is the distance from its tail to its tip. It is always a non-negative number, and it answers “how long is this arrow?” regardless of which direction it points.

Magnitude strips away direction and keeps only size. It is the foundation for unit vectors, distances, and the angle formula.

벡터의 크기(길이·노름, |a| 또는 ‖a‖)는 시작점에서 끝점까지의 거리이며, 항상 0 이상의 수입니다. 방향과 무관하게 "이 화살표가 얼마나 긴가"에 답합니다. 단위벡터·거리·각도 공식의 토대가 됩니다.

2. The Formula & Worked Examples

Square every component, add them, and take the square root:

|a| = √(a₁² + a₂² + … + aₙ²)

Worked example (3D). For a = (2, 3, 6):

|a| = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7

Worked example (2D). For a = (3, 4): |a| = √(9 + 16) = √25 = 5.

각 성분을 제곱해 더한 뒤 제곱근을 취합니다: |a| = √(a₁² + … + aₙ²). 예: (2, 3, 6) → √49 = 7, (3, 4) → √25 = 5.

3. Why It Is the Pythagorean Theorem

In 2D the components form the legs of a right triangle, and the vector is the hypotenuse — so |a| = √(a₁² + a₂²) is literally the Pythagorean theorem. In 3D and beyond, the same idea applies one axis at a time.

There is also a clean link to the dot product: a vector dotted with itself gives its length squared, |a|² = a · a. For a = (2, 3, 6), a · a = 4 + 9 + 36 = 49, so |a| = √49 = 7 — the same answer.

2D에서 성분은 직각삼각형의 두 변, 벡터는 빗변이므로 |a| = √(a₁² + a₂²)는 곧 피타고라스 정리입니다. 또한 자기내적과 연결됩니다: |a|² = a · a. 예: (2,3,6)의 a·a = 49 → |a| = 7.

4. Key Properties

• Non-negative. |a| ≥ 0, and |a| = 0 only for the zero vector.
• Scaling. |k a| = |k| |a| — multiplying by a scalar scales length by its absolute value.
• Length squared = self dot. |a|² = a · a.
• Triangle inequality. |a + b| ≤ |a| + |b| — a detour is never shorter than the direct path.

크기는 0 이상이며 영벡터일 때만 0입니다. |k a| = |k| |a|, |a|² = a · a, 삼각부등식 |a + b| ≤ |a| + |b| 가 성립합니다.

5. Frequently Asked Questions

How do you find the magnitude of a vector? Square each component, add them up, and take the square root: |a| = √(a₁² + … + aₙ²).

Can a magnitude be negative? No. It is a length, so it is always zero or positive; only the zero vector has magnitude 0.

What is the difference between magnitude and the dot product? Magnitude is a single vector’s length; the dot product combines two vectors. They connect through |a|² = a · a.

각 성분을 제곱·합산한 뒤 제곱근을 취합니다. 크기는 음수가 될 수 없으며 영벡터만 0입니다. 크기는 한 벡터의 길이, 내적은 두 벡터의 연산이며 |a|² = a · a 로 연결됩니다.

Ready to practice? Drill magnitude and the rest of linear algebra on C:Vector, or review the full vector reference and the related unit vector, dot product, and scalar multiplication.

실전 연습은 C:Vector에서, 전체 개념은 벡터 레퍼런스와 단위벡터, 내적, 스칼라배 문서에서 이어집니다.