45° angle
- Input
- A = (2, 1), B = (1, 3)
- Expected output
- A · B = 5; cos θ = √2/2; θ = 45°
|A| = √5, |B| = √10.
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dot product and angle between two vectors
Type A and B in Vector operations. The Dot product card shows the work, the Angle card gives the exact cosine, the value in degrees, in radians and in degrees, minutes and seconds, and the Relation card says whether they are orthogonal or parallel.
|A| = √5, |B| = √10.
Positive product.
Coordinate axes.
cos θ = -1.
68°33′45″.
B = 2·A.
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.
No: by definition it lies between 0° and 180°, the smaller angle between the directions. The direction angle of a single vector, measured from the x-axis, runs from 0° to 360° and appears on the Direction card.
The dot product is 0, but the angle is undefined because the zero vector has no direction and the formula would divide by zero. The calculator marks the angle and relation as undefined and says which vector is zero.
No. The dot product returns a number and measures how much the vectors point the same way (it uses cos θ). The cross product, defined in 3D only, returns a vector perpendicular to both and its magnitude measures the parallelogram area (it uses sin θ).
(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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