プラン

単位ベクトル計算機

この計算機はベクトルをその大きさで割り、同じ向きの単位ベクトルを求めます。たとえば ⟨3, 4⟩ は ⟨3/5, 4/5⟩ になります。2 次元・3 次元・4 次元のベクトルに対応し、正確な答え（根号は簡単にした形）を示し、結果の長さが 1 であることを確認します。

成分をカンマで区切って入力 — 2、3、または 4 個

結果

値を入力して「計算」を押すと結果が表示されます。

公式

v^=v∥v∥\hat{\mathbf{v}} = \frac{\mathbf{v}}{\lVert \mathbf{v} \rVert}

Divide each component by the magnitude of the vector.

∥⟨a,b,c⟩∥=a2+b2+c2\lVert \langle a, b, c \rangle \rVert = \sqrt{a^2 + b^2 + c^2}

The magnitude (length) of a vector, by Pythagoras.

この計算ツールの使い方

  1. Type the components of the vector separated by commas, such as 3, 4 or 1, 2, 2. Fractions and decimals work too.
  2. Press Calculate to get the unit vector, its decimals, and the magnitude of the original vector.
  3. The steps show the magnitude found by Pythagoras, then each component divided by it.

解き方付き例題

Easy: a 2D vector

|v| = √(3² + 4²) = √25 = 5.

Divide each component by 5: ⟨3/5, 4/5⟩.

Medium: a 3D vector

|v| = √(1 + 4 + 4) = 3.

The unit vector is ⟨1/3, 2/3, 2/3⟩.

Hard: a square root in the magnitude

|v| = √(4 + 1) = √5.

Divide and rationalise: 2/√5 = 2√5/5 and −1/√5 = −√5/5.

Unit vector ≈ ⟨0.894427, −0.447214⟩.

よくある間違い

  • Forgetting to square the negative components: (−1)² is 1, not −1.
  • Dividing by the sum of the components instead of the magnitude.
  • Trying to normalise the zero vector — it has no direction, so it has no unit vector.
  • Rounding the magnitude before dividing; keep √5 exact until the end.

よくある質問

What is a unit vector?

A vector of length 1. It keeps only the direction of the original vector, which is why it is used to describe directions.

How do I find the unit vector of a vector?

Find its magnitude with Pythagoras, √(a² + b² + c²), then divide every component by that magnitude.

How do I check a unit vector?

Square its components and add them: the total must be exactly 1. The calculator does this check automatically.