Result
Step-by-Step Calculation
| Vector | X | Y | Magnitude |
|---|---|---|---|
| A | 3 | 4 | 5 |
| B | 1 | 2 | 2.2361 |
| Result | 4 | 6 | 7.2111 |
Examples
Example 1: Addition
A = (3, 4)
B = (1, 2)
A + B = (4, 6)
Example 2: Dot Product
A = (1, 3, -5)
B = (4, -2, -1)
A · B = 3
Example 3: Magnitude
A = (3, 4, 5)
|A| = 7.0711
Example 4: Cross Product
A = (2, 3, 4)
B = (5, 6, 7)
A × B = (-3, 6, -3)
Vector Formulas
A ± B = (Ax ± Bx, Ay ± By, Az ± Bz)
|A| = √(Ax² + Ay² + Az²)
A · B = (Ax × Bx) + (Ay × By) + (Az × Bz)
A × B =
x: (AyBz - AzBy)
y: (AzBx - AxBz)
z: (AxBy - AyBx)
θ = cos⁻¹( (A · B) / (|A| |B|) )
projBA = ( (A · B) / |B|² ) B
Calculation History
No history yet.
About Vector Calculator & FAQ
Welcome to Vector Calculator, a comprehensive online mathematical tool designed for students, educators, engineers, and programmers. This tool accurately calculates both 2D and 3D vector operations without needing to install any software.
Vectors are fundamental entities in mathematics, physics, computer graphics, robotics, and engineering. They possess both magnitude (length) and direction, making them crucial for representing forces, velocities, and positional offsets.
Frequently Asked Questions
A vector is a mathematical object that has both a magnitude (length) and a direction. In a Cartesian coordinate system, it is represented by components (x, y for 2D, and x, y, z for 3D).
The magnitude (or length) of a vector A(x, y, z) is calculated using the Pythagorean theorem: |A| = √(x² + y² + z²).
The dot product (scalar product) takes two equal-length sequences of numbers and returns a single number. It is calculated as A·B = (Ax * Bx) + (Ay * By) + (Az * Bz). It's useful for finding the angle between vectors.
The cross product is a binary operation on two vectors in 3D space. It results in a third vector that is perpendicular to both of the input vectors. It is only valid for 3-dimensional vectors.
A unit vector is a vector with a magnitude of 1. It is used to specify a direction. You find it by dividing a vector by its own magnitude: A / |A|.
The angle θ is calculated using the dot product formula: cos(θ) = (A·B) / (|A|*|B|). You then use the inverse cosine (arccos) to find the angle in radians or degrees.
Yes, simply toggle the "3D" option at the top of the calculator. This will reveal the 'Z' input fields and enable 3D-specific operations like the Cross Product.
Vector projection gives the orthogonal projection of one vector onto a straight line parallel to another vector. Think of it as the "shadow" of vector A cast onto vector B.