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Also tell me if they're orthogonal or parallel.","exampleUseCases":[{"title":"Physics force projection problem","prompt":"I need to find the projection of force vector u = 3,4,0 onto direction vector v = 1,0,0, and also tell me if they're orthogonal."},{"title":"3D geometry angle check","prompt":"Can you compute the angle in degrees between vectors u = 1,0,0 and v = 0,1,0, and give me their dot product and cross product?"},{"title":"Computer graphics unit vector and distance","prompt":"For vectors u = 2,2,2 and v = 1,3,5, give me the unit vectors for each, the distance between their tips, and the parallelogram area they span."}],"resultDescription":"Returns a JSON object with dot product, cross product (if 3D), magnitudes of both vectors, angle in degrees and radians, cosine of angle, unit vectors, projection and scalar projection of u onto v, rejection of u from v, vector sum and difference, distance between tips, parallelogram area, triangle area, and boolean flags for whether vectors are parallel or orthogonal.","failureModes":["Vectors u and v have different lengths — returns error","Vector with fewer than 2 or more than 10 components — returns error","Cross product only defined for 3D vectors — field returns null for other dimensions","Zero vector input causes division-by-zero for unit vector or angle — may return null fields","Malformed query string (non-numeric components) — returns parse error"],"whenToPreferThis":"Choose this endpoint when you need a comprehensive suite of vector operation results in a single call — including dot product, cross product, angle, projections, unit vectors, and geometric area — rather than calling separate specialized endpoints. 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