THE NUMORIX GUIDE
How to use the Vector Calculator
Last reviewed September 14, 2026
What this calculator does
The operation selector computes 2D magnitude sqrt(x^2+y^2), direction atan2(y,x) in degrees, dot product x1*x2+y1*y2, or angle arccos((v1 dot v2)/(|v1||v2|)) in degrees.
Formula and method
The operation selector computes 2D magnitude sqrt(x^2+y^2), direction atan2(y,x) in degrees, dot product x1*x2+y1*y2, or angle arccos((v1 dot v2)/(|v1||v2|)) in degrees.
Variables and inputs
Mode defaults to Magnitude. Vector 1 defaults to (3,4); Vector 2 defaults to (0,0) and is needed only for dot product or angle mode.
Worked example
For magnitude of (3,4): |v| = sqrt(3^2+4^2) = sqrt(25) = 5. For v1=(3,4) and v2=(1,2), dot product = 3*1+4*2 = 11.
How to interpret the result
Magnitude is length, direction is the counterclockwise angle from the positive x-axis, dot product measures alignment, and angle mode returns the smaller angle between nonzero vectors.
Common mistakes to avoid
Do not use the dot product as a magnitude, and do not forget that angle requires two nonzero vectors. The direction result is in degrees even though atan2 is evaluated in radians.
Assumptions and limitations
Only two-dimensional vectors are supported. The angle branch clamps the cosine to [-1,1] for floating-point safety but does not provide a useful answer for a zero vector.
Practical use and checks
The Vector Calculator works with two-dimensional components and can return a magnitude, direction, dot product, or angle between vectors. Enter v1 = (3, 4) and choose magnitude as a check; the result should be 5. For v1 = (1, 0) and v2 = (0, 1), choose dot product to get 0 and angle to get 90 degrees. A direction result for (1, 1) should be 45 degrees, measured counterclockwise from the positive x-axis by atan2. Use magnitude for a resultant size, direction for orientation, dot product for projection or perpendicularity, and the angle for geometric comparison. Components must use the same coordinate system and units before they are combined. A zero vector has no unique direction and cannot define an angle with another vector; the engine may produce a nonfinite or guarded value in that edge case. Dot product zero indicates orthogonality in ordinary Euclidean coordinates, not that the vectors are absent. The page is two-dimensional and does not handle cross products, coordinate transforms, uncertainty, or time-varying components. For a physical force or velocity, preserve the sign of each component rather than entering magnitudes only, because direction is part of the vector's meaning.