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Vector Addition Calculator — Magnitude & Direction

Add two vectors by magnitude and angle. Decompose into X/Y components and find the resultant vector magnitude and direction.

Vector Addition (A + B)

Vector A

Vector B

Resultant Vector

14.1421
at 45°

Components

XY
A100
B010
R1010
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What is Vector Addition by Component Decomposition?

Vectors represent quantities that have both magnitude and direction, such as force, velocity, and displacement. You cannot simply add their magnitudes. Instead, you decompose each vector into perpendicular components (x and y), add the components separately, and then recombine them into a single resultant vector using the Pythagorean theorem and inverse tangent.

Mathematical Foundation

Component Decomposition
Ax=∣A∣cos⁡(θA),Ay=∣A∣sin⁡(θA)A_x = |A| \cos(\theta_A), \quad A_y = |A| \sin(\theta_A)
AxA_x= Horizontal (x) component of vector A.
AyA_y= Vertical (y) component of vector A.
∣A∣|A|= Magnitude (length) of vector A.
θA\theta_A= Direction angle of A measured from the positive x-axis.
Resultant Vector
∣R∣=Rx2+Ry2,θR=arctan⁡(RyRx)|R| = \sqrt{R_x^2 + R_y^2}, \quad \theta_R = \arctan\left(\frac{R_y}{R_x}\right)
∣R∣|R|= Magnitude of the resultant vector.
RxR_x= Sum of all x-components: Ax + Bx.
RyR_y= Sum of all y-components: Ay + By.
θR\theta_R= Direction angle of the resultant.

Laws & Principles

  • Component Independence: The x-components and y-components are independent. Changes to horizontal motion do not affect vertical motion. This is why a ball thrown horizontally falls at the same rate as one dropped straight down.
  • Angle Convention: Angles are measured counterclockwise from the positive x-axis. 0 degrees points right, 90 degrees points up, 180 degrees points left, 270 degrees points down.
  • Triangle Inequality: The resultant magnitude is always less than or equal to the sum of the individual magnitudes. Equality holds only when both vectors point in exactly the same direction.

Step-by-Step Example Walkthrough

" A hiker walks 5 km at 30 degrees north of east, then 3 km at 120 degrees (northwest). Find the total displacement. "

  1. 1. Vector A components: Ax = 5*cos(30) = 4.33 km, Ay = 5*sin(30) = 2.50 km.
  2. 2. Vector B components: Bx = 3*cos(120) = -1.50 km, By = 3*sin(120) = 2.60 km.
  3. 3. Sum components: Rx = 4.33 + (-1.50) = 2.83 km, Ry = 2.50 + 2.60 = 5.10 km.
  4. 4. Resultant magnitude: |R| = sqrt(2.83^2 + 5.10^2) = 5.83 km.
  5. 5. Resultant angle: theta = arctan(5.10 / 2.83) = 61.0 degrees.
Final Result: The hiker ends up 5.83 km from the starting point, at 61.0 degrees north of east.

Quick Answer: How do you add two vectors?

Break each vector into its x and y components using cosine and sine. Add all the x-components together to get Rx. Add all the y-components to get Ry. The resultant magnitude is sqrt(Rx^2 + Ry^2) and the direction is arctan(Ry/Rx).

The Component Method

Rx = Ax + Bx, Ry = Ay + By, |R| = sqrt(Rx^2 + Ry^2)

This method works for any number of vectors, not just two. For three vectors, add Cx to Rx and Cy to Ry. The component method is always exact, unlike graphical (tip-to-tail) methods which depend on drawing precision.

Key Angle Components

Angle cos(theta) sin(theta) Direction
01.0000.000East (right)
300.8660.500NNE
450.7070.707NE (diagonal)
900.0001.000North (up)
180-1.0000.000West (left)

Practical Applications

River Crossing

A boat heading north at 5 m/s crosses a river with a 3 m/s eastward current. The actual velocity is the vector sum: magnitude = sqrt(25+9) = 5.83 m/s at arctan(3/5) = 31 degrees east of north. The boat drifts downstream unless it adjusts its heading.

Force Equilibrium

Two ropes hold a 100 N weight. If rope A pulls at 120 degrees and rope B pulls at 60 degrees, the vector sum of their tensions must equal 100 N downward (at 270 degrees) for the system to remain stationary.

Pro Tips

Do This

  • ✓Check the quadrant of your resultant angle. arctan only returns values from -90 to 90 degrees. If Rx is negative, you must add 180 degrees. If both Rx and Ry are negative, the resultant is in the third quadrant.
  • ✓Sketch a rough diagram first. A quick sketch confirms whether your resultant direction makes physical sense. If two forces pull mostly upward, the resultant should point up, not down.

Avoid This

  • ✗Do not add magnitudes directly. A 3 N force at 0 degrees plus a 4 N force at 90 degrees does not equal 7 N. It equals 5 N at 53.1 degrees (the 3-4-5 triangle).
  • ✗Do not mix radians and degrees. Most scientific calculators default to radians. cos(90 radians) is not 0. cos(90 degrees) is 0. Always verify your angle mode before computing.

Frequently Asked Questions

Can two vectors of different magnitudes cancel each other out?

Only if they have equal magnitudes and opposite directions (180 degrees apart). A 5 N force east plus a 5 N force west equals zero. A 5 N force east plus a 3 N force west does not cancel; it produces a net 2 N force eastward.

What is the difference between vector addition and the dot product?

Vector addition produces a new vector (the resultant). The dot product produces a scalar (a single number). The dot product measures how much two vectors point in the same direction. Two perpendicular vectors have a dot product of zero. Vector addition combines forces or velocities; the dot product calculates work done by a force.

Why do I need to use arctan2 instead of arctan?

Standard arctan(y/x) cannot distinguish between quadrants. arctan(1/1) and arctan(-1/-1) both return 45 degrees, but the first vector points northeast (Q1) and the second points southwest (Q3). The atan2(y, x) function uses both signs to return the correct angle from 0 to 360 degrees without manual quadrant adjustment.

Does vector addition work in three dimensions?

Yes. In 3D, each vector has three components (x, y, z). You add all x-components, all y-components, and all z-components separately. The resultant magnitude is sqrt(Rx^2 + Ry^2 + Rz^2). Direction requires two angles (azimuth and elevation) instead of one. This calculator handles 2D only.

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