Add a 10N east vector and an 8N north vector. What is the resultant magnitude and angle?

R = √(10²+8²) = √164 ≈ 12.8N; θ = arctan(8/10) ≈ 38.7° north of east.

Three forces balance (net = 0). If two are known, how do you find the third?

F₃ = −(F₁ + F₂); add F₁ and F₂ first, then negate the resultant.

A river flows east at 3 m/s. A swimmer swims north at 4 m/s relative to the water. What is their speed relative to the ground?

V_ground = √(3²+4²) = 5 m/s at arctan(3/4)=37° east of north.

Pro 🔒~15 min

Vector Addition

Add vectors graphically and by components

How it works

Vectors have both magnitude and direction. Vector addition follows the parallelogram law (or tip-to-tail): place vectors head to tail and draw the resultant from start to end. Equivalently, add components separately: R_x = A_x + B_x, R_y = A_y + B_y. The magnitude and direction of the resultant follow from Pythagorean theorem and arctangent. This is the foundation for analyzing 2D forces, velocity, and displacement.

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Step-by-step

Drag the vector arrows to set magnitude and direction.
The resultant vector (green) appears automatically.
Switch between graphical and component modes.
Verify that tip-to-tail gives the same answer as component addition.

Key formulas

$\vec{R} = \vec{A} + \vec{B}$ Vector sum (resultant)
$R_x = A_x + B_x = A\cos\theta_A + B\cos\theta_B$ x-components
$R = \sqrt{R_x^2 + R_y^2}$ Resultant magnitude

Frequently asked questions

Add a 10N east vector and an 8N north vector. What is the resultant magnitude and angle?: R = √(10²+8²) = √164 ≈ 12.8N; θ = arctan(8/10) ≈ 38.7° north of east.
Three forces balance (net = 0). If two are known, how do you find the third?: F₃ = −(F₁ + F₂); add F₁ and F₂ first, then negate the resultant.
A river flows east at 3 m/s. A swimmer swims north at 4 m/s relative to the water. What is their speed relative to the ground?: V_ground = √(3²+4²) = 5 m/s at arctan(3/4)=37° east of north.