Sum and difference in the plane
- Input
- A = (3, 4), B = (-1, 2)
- Expected output
- A + B = (2, 6); A - B = (4, 2)
|A| = 5, |B| = √5.
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vector addition calculator
In Vector operations, choose 2D or 3D and type A and B. The cards show A + B, A - B, k·A, the magnitudes and the unit vectors, and the drawing switches between parallelogram, head to tail and difference.
|A| = 5, |B| = √5.
Same direction, magnitude 10.
|A| = √14.
Right angles: exact result.
Zero sum.
Magnitude 1.
Add or subtract component by component: A + B = (a₁ + b₁, a₂ + b₂) and, in 3D, also a₃ + b₃. For A = (3, 4) and B = (-1, 2): A + B = (2, 6) and A - B = (4, 2). Geometrically, the sum is the diagonal of the parallelogram built on A and B, or the vector that closes the chain when you place the tail of B on the tip of A.
Only when the vectors point the same way. In general the triangle inequality holds, |A + B| ≤ |A| + |B|. With A = (3, 4) and B = (-1, 2): |A + B| = 2√10 ≈ 6.32, while |A| + |B| ≈ 7.24.
A point marks a position; a vector marks a displacement (direction, sense and magnitude). Here vectors are drawn from the origin, so the vector (3, 4) ends at the point (3, 4). The vector between two points A and B is B - A.
Yes: add the result to the next one, or use the Resultant of forces mode, which accepts up to 8 vectors in the plane given by magnitude and angle and draws the head-to-tail polygon.
(2, 6)
(4, 2)
(6, 8)
5
√5
≈ 2.2361
(3/5, 4/5)
≈ (0.6, 0.8)
(-√5/5, 2√5/5)
≈ (-0.4472, 0.8944)
53.1301°
5
63.4349°
Neither parallel nor orthogonal
10
10
5
(-1, 2)
√5
≈ 2.2361
(4, 2)
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