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Resultant of Two Vectors at an Angle

R=A2+B2+2ABcos⁡θR = \sqrt{A^2 + B^2 + 2AB\cos\theta}
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Worked example: 3 and 4 at right angles → resultant 5. Press Try an example to run it live, then adjust anything.

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Dot Product from Magnitudes and Included AngleCross Product Magnitudex-Component from Magnitude and Angley-Component from Magnitude and Angle
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Cross Product Magnitude: Both use resultant magnitude and angle between vectors.Dot Product from Magnitudes and Included Angle: Both use resultant magnitude and angle between vectors.Angle Between Two 2D Vectors (Components): Both use angle between vectors.Magnitude of a 2D Vector: Both use resultant magnitude.Magnitude of a 3D Vector: Both use resultant magnitude.Magnitude of a Scaled Vector: Both use resultant magnitude.
4 more relatedScalar Projection of One Vector onto Another: Both use resultant magnitude.Unit Vector Component (Normalization): Both use resultant magnitude.x-Component from Magnitude and Angle: Both use resultant magnitude.y-Component from Magnitude and Angle: Both use resultant magnitude.
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Resultant of Two Vectors at an Angle explained

θABR

Two vectors add head to tail, and the closing side of the parallelogram they form is the resultant. Its length is √(A² + B² + 2AB cos θ), where θ is the angle between the two vectors laid tail to tail. At 90° the cosine term vanishes and the familiar √(A² + B²) returns: 3 and 4 at right angles give exactly 5. At 0° the vectors reinforce completely and R = A + B; at 180° they fight and R = |A − B|. Simon Stevin argued the rule into existence in 1586 with his chain of beads over an inclined plane, and Newton made it Corollary I of the Principia in 1687.

Watch the sign. The law of cosines carries a minus, this formula a plus, and the difference is not a typo: the law of cosines uses the interior angle of the triangle, while θ here is the angle between the vectors, and the two are supplementary. Mixing them up is the single most common error in force-addition problems. Worked backwards: two forces of 5 N and 8 N producing a 7 N resultant must be separated by arccos((49 − 25 − 64)/80) = arccos(−0.5) = 120°.

Resultant of Two Vectors at an Angle formula

R=A2+B2+2ABcos⁡θR = \sqrt{A^2 + B^2 + 2AB\cos\theta}
Where
  • RR= Resultant magnitude
  • AA= Magnitude of a
  • BB= Magnitude of b
  • θ\theta= Angle between a and b (°)
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Vector operations

12 formulasMagnitude, direction, components and unit vectors, plus the dot and cross products and the projections and angles they give you.

Revised 31 October 2026
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formula.expert. (2026, October 31). Resultant of Two Vectors at an Angle Formula & Calculator. https://formula.expert/solvers/vector-addition-resultant

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