Drag input vector arrowheads (v₁, v₂) or step through projection subtraction
The source projection factor is the dot product of the vector and basis divided by the squared basis length. For basis two comma zero, its squared length is four. Vectors two comma three and four comma three give dot products four and eight, so their factors are one and two. Bars share one hundred sixty pixels per factor unit. Near-zero squared basis length returns zero projection under the source numerical guard. With first basis two comma zero and second vector two comma three, projection is two comma zero. Subtracting it leaves zero comma three, orthogonal to the first basis. The second input length is square root thirteen, approximately three point six zero six; residual length is three. Bars share one hundred pixels per length unit. Length changes follow subtraction, not normalization. The source normalization divides each component by vector length when length exceeds its tiny numerical guard. Residual zero comma three therefore becomes zero comma one, reducing length three to one. Bars share one hundred pixels per length unit. The computed display basis depends on the selected algorithm step and normalization toggle. Full original JSON and genuine reset plus native responsive first-footer checks preserve the source tool.