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Parallelogram law
Beginner
Resultant is the diagonal of the parallelogram built on two vectors.
R² = A²+B²+2AB cosθTwo vectors from the same point form the sides of a parallelogram; the diagonal from that point is their resultant. Its magnitude is R = √(A² + B² + 2AB cosθ), with θ the angle between them. It is the same as head-to-tail addition.
Forces 3 N and 4 N at 90°: R = √(9+16) = 5 N. At 0° they give 7 N; at 180°, 1 N.
θ is the angle between the tails when both start at the same point, not the exterior angle.
In more detail
The parallelogram law says that two vectors starting from the same point are added by drawing a parallelogram. The resultant is the diagonal from the common starting point.
Real-world uses
- Two ropes pulling a box at an angle have a resultant force along the diagonal of their parallelogram.
- Wind and aircraft airspeed combine into ground speed using this construction.
- Two people pulling a sled at an angle find the resultant force using the parallelogram law, which gives the combined effect in one step.
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Frequently asked questions
Is the parallelogram law the same as head-to-tail?
Yes, the two methods give the same result. The parallelogram is a visual form of head-to-tail addition.
What if the vectors are parallel?
The resultant is simply the sum of their magnitudes along the same line.
Does the parallelogram law work for any angle?
Yes. The formula R² = A² + B² + 2AB cos θ applies for any angle θ between the two vectors.
Related topics in Measurement & Math Tools
Scalars and vectors · Vector representation · Vector addition · Vector components · Unit vectors · Dot product