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Vector addition
Beginner
Add vectors head-to-tail or by components.
R = A + BTo add vectors geometrically, place the tail of the second at the head of the first; the sum runs from the first tail to the last head. Algebraically, add like components: (Ax+Bx, Ay+By). Addition is commutative and associative.
A = (3, 1), B = (2, 4): A + B = (5, 5), magnitude √50 ≈ 7.07.
Magnitudes do not add unless the vectors point the same way. |A + B| ≤ |A| + |B|.
In more detail
Vectors add head to tail: place the tail of the second vector at the head of the first, and the resultant runs from the start to the end. The order of addition does not change the result.
Real-world uses
- A boat's path combines its own velocity with the river current by vector addition.
- Walking directions combine several straight legs into one net displacement.
- A pilot adds the wind velocity to the aircraft velocity to find the actual ground velocity of the plane.
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Frequently asked questions
Does the order of adding vectors matter?
No. Vector addition is commutative, so A + B equals B + A.
How do I add vectors that are not at right angles?
Resolve each vector into components along two axes, add the components, then find the magnitude and direction of the result.
How do I subtract two vectors?
Add the negative of the second vector, which reverses its direction, then use the same head-to-tail rule.
Related topics in Measurement & Math Tools
Unit conversion · Scalars and vectors · Vector representation · Parallelogram law · Vector components · Unit vectors