Resultant of Two Vectors at an Angle
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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 magnitude
- = Magnitude of a
- = Magnitude of b
- = Angle between a and b
- Resultant magnitude — Dot Product from Magnitudes and Included Angle, Cross Product Magnitude
- Magnitude of a — Dot Product from Magnitudes and Included Angle, Cross Product Magnitude
- Magnitude of b — Dot Product from Magnitudes and Included Angle, Cross Product Magnitude
- Angle between a and b — Dot Product from Magnitudes and Included Angle, Cross Product Magnitude