Vector Calculator
Perform 2D and 3D vector operations step-by-step: dot product, cross product, magnitude, angle, addition, and subtraction.
2D & 3D Vector Operations Engine
Select vector mode and enter component coordinates.
Vector Operations Output
Quick Summary
Our vector calculator 2D 3D dot cross product magnitude steps tool computes comprehensive vector math operations. It calculates dot product $\mathbf{A} \cdot \mathbf{B}$, cross product $\mathbf{A} \times \mathbf{B}$, Euclidean magnitudes $\|\mathbf{A}\|, \|\mathbf{B}\|$, angle between vectors $\theta$, component addition $\mathbf{A} + \mathbf{B}$, and subtraction $\mathbf{A} - \mathbf{B}$.
How It Works: Vector Algebra & Geometry
Vectors represent quantities with both magnitude and spatial direction:
1. **Vector Magnitude (Length):** $\|\mathbf{A}\| = \sqrt{a_x^2 + a_y^2 + a_z^2}$.
2. **Dot Product (Scalar Product):** $\mathbf{A} \cdot \mathbf{B} = a_x b_x + a_y b_y + a_z b_z = \|\mathbf{A}\| \|\mathbf{B}\| \cos\theta$.
3. **Cross Product (Vector Product):** $\mathbf{A} \times \mathbf{B} = \langle a_y b_z - a_z b_y, a_z b_x - a_x b_z, a_x b_y - a_y b_x \rangle$ (perpendicular to both vectors!).
4. **Angle Between Vectors:** $\theta = \arccos\left(\frac{\mathbf{A} \cdot \mathbf{B}}{\|\mathbf{A}\| \|\mathbf{B}\|}\right)$.
5. **Component Addition/Subtraction:** Sum $a_i + b_i$, difference $a_i - b_i$.
Formula Explanation
Your 2D and 3D vector calculations follow fundamental vector space rules:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for 3D vectors $\mathbf{A} = \langle 3, 4, 0 \rangle$ and $\mathbf{B} = \langle 1, 2, 2 \rangle$:
- Step 1 (Calculate Magnitudes): $\|\mathbf{A}\| = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{25} = \mathbf{5}$. $\|\mathbf{B}\| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = \mathbf{3}$.
- Step 2 (Calculate Dot Product): $\mathbf{A} \cdot \mathbf{B} = (3)(1) + (4)(2) + (0)(2) = 3 + 8 + 0 = \mathbf{11}$.
- Step 3 (Calculate Angle θ): $\cos\theta = \11 / (5)(3) = \11 / 15 \approx 0.7333 \implies \theta = \arccos(0.7333) \approx \mathbf{42.83^\circ}$.
- Step 4 (Calculate Cross Product A × B): $\langle (4)(2)-(0)(2), (0)(1)-(3)(2), (3)(2)-(4)(1) \rangle = \mathbf{\langle 8, -6, 2 \rangle}$.
- Step 5 (Compute Sum A + B): $\mathbf{A} + \mathbf{B} = \langle 3+1, 4+2, 0+2 \rangle = \mathbf{\langle 4, 6, 2 \rangle}$, delivering **A · B = 11, A × B = <8, -6, 2>**!
Calculation Examples: Real-World Scenario Comparison
Compare vector modes, inputs, dot products, cross products, and angles:
| Vector A | Vector B | Dot Product (A · B) | Cross Product (A × B) | Angle θ |
|---|---|---|---|---|
| ⟨3, 4, 0⟩ | ⟨1, 2, 2⟩ | 11 | ⟨8, -6, 2⟩ | 42.83° |
| ⟨3, 4⟨ | ⟨-4, 3⟨ | 0 (Orthogonal / Perpendicular) | ⟨0, 0, 25⟩ | 90.00° |
| ⟨2, 4, 6⟩ | ⟨1, 2, 3⟩ | 28 | ⟨0, 0, 0⟩ (Parallel) | 0.00° |
| ⟨5, 0⟨ | ⟨-5, 0⟨ | -25 (Opposing Directions) | ⟨0, 0, 0⟩ | 180.00° |
Benefits of Using the Vector Calculator
Utilizing this calculator provides essential physics force mechanics, 3D graphics rendering, and navigation advantages:
- Physics Force & Work Calculations ($W = \mathbf{F} \cdot \mathbf{d}$): Computes work done by force vectors applied over displacement vectors.
- Torque & Angular Momentum ($\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$): Calculates rotational torque perpendicular vectors.
- 3D Graphics Surface Normal Vectors: Computes polygon lighting normals using cross products of mesh edge vectors.
- Navigation & Flight Path Dynamics: Combines wind velocity vectors and aircraft heading vectors to compute ground velocity.
Frequently Asked Questions (FAQ)
What is a vector?
A vector is a mathematical quantity that has both magnitude (length) and direction, represented by components (x, y) in 2D or (x, y, z) in 3D.
What is the difference between dot product and cross product?
The dot product results in a scalar (single number); the cross product results in a new vector that is perpendicular to both original vectors.
What does a dot product of zero mean?
If A · B = 0 for non-zero vectors, the vectors are orthogonal (perpendicular at 90°).
What does a cross product of zero vector mean?
If A × B = ⟨0, 0, 0⟩, the vectors are parallel (0°) or anti-parallel (180°).
How do you calculate vector magnitude ||A||?
Using the Pythagorean distance formula: ||A|| = sqrt(x^2 + y^2 + z^2).
Is cross product commutative (A × B = B × A)?
NO. The cross product is anti-commutative: A × B = -(B × A).
Is dot product commutative (A · B = B · A)?
YES. A · B = B · A.
How do you find a unit vector in direction of A?
Divide each component of A by its magnitude: u = A / ||A||.
What is the geometric meaning of cross product magnitude ||A × B||?
||A × B|| equals the area of the parallelogram formed by vectors A and B.
How is cross product used in 3D game engines?
Game engines use cross products to determine camera orientation vectors (up, right, look-at) and surface lighting normals.