प्लान

इकाई सदिश कैलकुलेटर

यह कैलकुलेटर सदिश को उसके परिमाण से भाग देकर उसी दिशा में इकाई सदिश देता है — जैसे ⟨3, 4⟩ बन जाता है ⟨3/5, 4/5⟩। यह 2D, 3D और 4D सदिशों के लिए काम करता है, उत्तर सटीक रखता है (सरल करणियों के साथ) और जाँचता है कि परिणाम की लंबाई 1 है।

घटक अल्पविराम से अलग — 2, 3 या 4

नतीजा

मान डालें और नतीजा देखने के लिए “गणना करें” दबाएँ।

सूत्र

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.

यह कैलकुलेटर कैसे इस्तेमाल करें

  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.

हल किए गए उदाहरण

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⟩.

आम गलतियाँ

  • 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.

अक्सर पूछे जाने वाले सवाल

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.