How this calculator works
An orthogonal projection splits A into a component parallel to B and a perpendicular remainder. The projection factor scales B into that parallel component. The scalar projection measures signed length along B's direction, so it can be negative. A projection alone cannot recover all of A: any perpendicular remainder would give the same projection. If every other component of A is known, one missing component can be recovered when the corresponding component of B is nonzero. A known projection and nonzero factor also recover B.
Formula
proj_B(A) = ((A dot B)/(B dot B)) B. Factor = (A dot B)/|B|^2. Scalar projection = (A dot B)/|B|. Rejection = A - proj_B(A).
Worked example
A = (3, 4) projected onto B = (2, 0) gives factor 6/4 = 1.5, vector (3, 0), signed length 3 and rejection (0, 4). Projecting (-2, 3, 4) onto (1, 0, 0) instead gives (-2, 0, 0).
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Frequently asked questions
Does reversing B change the projected vector?
No. Its span is the same line, so the projected vector stays the same. The scalar projection changes sign because B's positive direction reverses.
What if A and B are perpendicular?
Their dot product is zero, making the projection vector zero and leaving A entirely in the rejection.
Why can B not be zero?
A zero vector defines no direction and makes the denominator zero. A nonzero B is required in every mode.