Orthogonal Vectors Calculator

Two vectors are orthogonal if their dot product equals zero. This means the angle between them is exactly 90 degrees. The dot product a . b = axbx + ayby + az*bz, so checking orthogonality is a single arithmetic test.

The angle between any two non-zero vectors is given by the dot product formula: cos(theta) = (a . b) / (|a| * |b|), where |a| is the magnitude of a. Rearranging gives theta = arccos(a . b / (|a| * |b|)). The angle is in degrees here.

The cross product a x b gives a vector perpendicular to both a and b. Its magnitude is |a|*|b|*sin(theta), which equals the area of the parallelogram formed by a and b. If the cross product is the zero vector, the vectors are parallel (or one is the zero vector).

Gram-Schmidt orthogonalization takes b and removes its component along a, producing b_perp = b - ((a . b) / (a . a)) * a. This vector is guaranteed to be orthogonal to a. If b is already orthogonal to a, b_perp equals b. Gram-Schmidt is the basis of QR decomposition and the construction of orthogonal bases in linear algebra.

Orthogonality is central to physics (forces at right angles do no work on each other), computer graphics (camera coordinate systems use three mutually orthogonal vectors), signal processing (orthogonal waveforms do not interfere), and statistics (orthogonal contrasts in ANOVA can be tested independently).

In machine learning, principal components are orthogonal by construction. Each PC captures variation that is uncorrelated with every other PC, which is why the principal components decompose variance cleanly without overlap.