الخطط

حاسبة متجه الوحدة

تقسم هذه الآلة الحاسبة المتجه على مقداره لتعطي متجه الوحدة في الاتجاه نفسه — مثلًا يصبح ⟨3, 4⟩ هو ⟨3/5, 4/5⟩. تعمل مع المتجهات ثنائية وثلاثية ورباعية الأبعاد، وتُبقي الإجابات دقيقة (مع تبسيط الجذور)، وتتحقق من أن طول الناتج 1.

مركبات مفصولة بفواصل — 2 أو 3 أو 4

النتيجة

أدخل القيم واضغط احسب لرؤية النتيجة.

الصيغة

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.

كيفية استخدام هذه الحاسبة

  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.

أمثلة محلولة

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⟩.

أخطاء شائعة

  • 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.

الأسئلة الشائعة

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.