# Vektör hesaplayıcı

> Dot and cross products of two vectors in 2D or 3D, their magnitudes, the angle between them and the projection of one onto the other, with a drawing.

Etkileşimli sürüm: https://www.calcopenly.com/tr/math/vector-calculator
Konu: Matematik hesaplayıcıları

The dot product multiplies matching components and adds them, a · b = a₁b₁ + a₂b₂ + a₃b₃. The cross product of two 3D vectors is a vector perpendicular to both, and its length equals the area of the parallelogram they span. The angle between the vectors is computed as θ = atan2(|a × b|, a · b), which equals arccos(a · b / |a||b|) but stays accurate for nearly parallel vectors, and the projection of a onto b is (a · b / |b|²) b.

Physics (work and torque), 3D graphics and linear algebra are the usual uses. The default vectors a = (1, 2, 3) and b = (4, 5, 6) give a · b = 32, a × b = (−3, 6, −3) and an angle of about 12.93°.

A dot product of 0 means the vectors are perpendicular; a zero cross product means they are parallel or one is zero. In 2D mode both vectors lie in the xy-plane, so their cross product points along z: (3, 4) × (4, −3) = (0, 0, −25).

## Girdiler

- **Dimensions** (seçenekler: 2D, 3D)
- **a — x**
- **a — y**
- **a — z**
- **b — x**
- **b — y**
- **b — z**

## Sonuçlar

- Dot product a · b — ana sonuç
- Cross product a × b
- |a × b| (parallelogram area)
- |a|
- |b|
- Angle between a and b (°)
- Angle in radians (rad)
- Scalar projection of a onto b
- Vector projection of a onto b

## Formül

$$
\begin{gathered} \mathbf a\cdot\mathbf b = \sum a_i b_i \\[6pt] \theta = \operatorname{atan2}(|\mathbf a\times\mathbf b|,\ \mathbf a\cdot\mathbf b) \\[6pt] \operatorname{proj}_{\mathbf b}\mathbf a = \frac{\mathbf a\cdot\mathbf b}{|\mathbf b|^2}\,\mathbf b \end{gathered}
$$

## Çözümlü örnekler

### a = (1, 2, 3), b = (4, 5, 6)

- Dimensions: 3D
- a — x: 1
- a — y: 2
- a — z: 3
- b — x: 4
- b — y: 5
- b — z: 6
- **Dot product a · b: 32**
- **Cross product a × b: (−3, 6, −3)**
- **|a|: 3.7416573868**
- **Angle between a and b: 12.93315449 °**
- **|a × b| (parallelogram area): 7.3484692283**
- Doğrulama kaynağı: Hand calculation; Python decimal √14, √54 and atan2(√54, 32) in degrees (hp.py)

### Perpendicular 2D vectors (3, 4) and (4, −3)

- Dimensions: 2D
- a — x: 3
- a — y: 4
- b — x: 4
- b — y: -3
- **Dot product a · b: 0**
- **Angle between a and b: 90 °**
- **Cross product a × b: (0, 0, −25)**
- Doğrulama kaynağı: 3·4 + 4·(−3) = 0; 3·(−3) − 4·4 = −25

### Parallel vectors (edge case)

- Dimensions: 3D
- a — x: 1
- a — y: 2
- a — z: 3
- b — x: 2
- b — y: 4
- b — z: 6
- **Cross product a × b: (0, 0, 0)**
- **Angle between a and b: 0 °**
- **Dot product a · b: 28**
- Doğrulama kaynağı: b = 2a, so a × b = 0 and θ = 0

### Projection of (2, 3) onto (4, 0)

- Dimensions: 2D
- a — x: 2
- a — y: 3
- b — x: 4
- b — y: 0
- **Scalar projection of a onto b: 2**
- **Vector projection of a onto b: (2, 0)**
- **Angle between a and b: 56.30993247 °**
- Doğrulama kaynağı: a·b/|b| = 8/4; angle atan2(3, 2) = 56.3099324740202…° (Python math.degrees)

### Opposite directions (1, 0) and (−2, 0)

- Dimensions: 2D
- a — x: 1
- a — y: 0
- b — x: -2
- b — y: 0
- **Angle between a and b: 180 °**
- **Dot product a · b: -2**
- **Scalar projection of a onto b: -1**
- Doğrulama kaynağı: Antiparallel vectors: θ = 180°, a·b = −2

## Sorular

### How do you calculate the dot product of two vectors?

Multiply corresponding components and add the results: a · b = a₁b₁ + a₂b₂ + a₃b₃. For (1, 2, 3) · (4, 5, 6) that is 4 + 10 + 18 = 32. The dot product also equals |a||b| cos θ, so it is positive when the angle is under 90°, 0 at exactly 90° and negative beyond it: (1, 0) · (−2, 0) = −2.

### How do you calculate the cross product?

For a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). For (1, 2, 3) × (4, 5, 6) that gives (12 − 15, 12 − 6, 5 − 8) = (−3, 6, −3). The result is perpendicular to both vectors, follows the right-hand rule, and changes sign if the order is swapped: b × a = (3, −6, 3).

### How do you find the angle between two vectors?

Use cos θ = (a · b)/(|a||b|). For (1, 2, 3) and (4, 5, 6), cos θ = 32/(√14 × √77) ≈ 0.974632, so θ ≈ 12.93°. Near 0° or 180° the arccos form loses accuracy, so the calculator uses the equivalent θ = atan2(|a × b|, a · b), the form William Kahan recommends in his notes on floating-point roundoff.

### What does it mean if the dot product is zero?

The two vectors are perpendicular (orthogonal), provided neither is the zero vector. (3, 4) · (4, −3) = 12 − 12 = 0, so those vectors meet at exactly 90°. In physics the same test shows that a force at right angles to the motion does no work, since work is the dot product of force and displacement.

### What is the difference between scalar and vector projection?

The scalar projection of a onto b is the signed length of a along b, a · b / |b|; the vector projection is that length times the unit vector of b, (a · b / |b|²) b. Projecting (2, 3) onto (4, 0) gives a scalar projection of 8/4 = 2 and a vector projection of (2, 0). A negative scalar projection means a points partly against b.

### “Vektör hesaplayıcı” ne kadar doğru sonuç verir?

Doğruluk, girdilerinize ve yöntemin varsayımlarına bağlıdır. Ondalık aritmetik 50 anlamlı basamak kullanır; ancak tahminler, sayısal yöntemler ve kaynak veriler daha az hassas olabilir. Gösterilen değerin yuvarlanması bu sınırları ortadan kaldırmaz. Bağımsız kaynaklarla doğrulanan çözümlü örnek sayısı: 5. Örneğin “a = (1, 2, 3), b = (4, 5, 6)”, Hand calculation; Python decimal √14, √54 and atan2(√54, 32) in degrees (hp.py) ile karşılaştırılarak doğrulanır.

### Yöntemin kaynağı nedir?

Wolfram MathWorld — Dot Product; Wolfram MathWorld — Cross Product; W. Kahan, How futile are mindless assessments of roundoff in floating-point computation? §12 — angles via atan2 rather than arccos.

## Kaynaklar

- [Wolfram MathWorld — Dot Product](https://mathworld.wolfram.com/DotProduct.html)
- [Wolfram MathWorld — Cross Product](https://mathworld.wolfram.com/CrossProduct.html)
- [W. Kahan, How futile are mindless assessments of roundoff in floating-point computation? §12 — angles via atan2 rather than arccos](https://people.eecs.berkeley.edu/~wkahan/Mindless.pdf)