All Tools

Vector Operations Visualizer

Enter two vectors by components or by magnitude and angle. Choose an operation to see the result visualized on a 2D coordinate plane with full step-by-step calculations.

Vector Diagram

xyABR

Parameters

Vector A

Vector B

Results

|A|
5
Angle of A
53 deg
|B|
2.236
Angle of B
63 deg
Angle between A and B
10 deg
Resultant
(4, 6)
|Resultant|
7.211
Resultant angle
56 deg

Step-by-Step Calculation

1. Magnitude of A

∣A⃗∣=Ax2+Ay2|\vec{A}| = \sqrt{A_x^2 + A_y^2}
∣A⃗∣=(3)2+(4)2|\vec{A}| = \sqrt{(3)^2 + (4)^2}
∣A⃗∣=5|\vec{A}| = 5

2. Magnitude of B

∣B⃗∣=Bx2+By2|\vec{B}| = \sqrt{B_x^2 + B_y^2}
∣B⃗∣=(1)2+(2)2|\vec{B}| = \sqrt{(1)^2 + (2)^2}
∣B⃗∣=2.236|\vec{B}| = 2.236

3. Vector Addition

R⃗=A⃗+B⃗=(Ax+Bx,  Ay+By)\vec{R} = \vec{A} + \vec{B} = (A_x + B_x,\; A_y + B_y)
R⃗=(3+1,  4+2)\vec{R} = (3 + 1,\; 4 + 2)
R⃗=(4,  6)\vec{R} = (4,\; 6)

4. Resultant Magnitude

∣R⃗∣=Rx2+Ry2|\vec{R}| = \sqrt{R_x^2 + R_y^2}
∣R⃗∣=(4)2+(6)2|\vec{R}| = \sqrt{(4)^2 + (6)^2}
∣R⃗∣=7.211|\vec{R}| = 7.211

Reference Guide

Vector Addition

To add two vectors, add their corresponding components. Graphically you can use the tip-to-tail method or the parallelogram method.

R⃗=A⃗+B⃗=(Ax+Bx,  Ay+By)\vec{R} = \vec{A} + \vec{B} = (A_x + B_x,\; A_y + B_y)

Dot Product

The dot product measures how much two vectors point in the same direction. It equals zero for perpendicular vectors.

A⃗⋅B⃗=∣A⃗∣∣B⃗∣cos⁡θ=AxBx+AyBy\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta = A_x B_x + A_y B_y

Cross Product (2D)

In 2D the cross product gives a single scalar that represents the z-component of the full 3D cross product. Its absolute value equals the area of the parallelogram formed by the two vectors.

A⃗×B⃗=AxBy−AyBx\vec{A} \times \vec{B} = A_x B_y - A_y B_x

Magnitude and Direction

Every 2D vector can be described by its magnitude (length) and the angle it makes with the positive x-axis.

∣A⃗∣=Ax2+Ay2,θ=arctan⁡AyAx|\vec{A}| = \sqrt{A_x^2 + A_y^2}, \quad \theta = \arctan\frac{A_y}{A_x}