How this instrument works
The scalar projection of vector v onto vector u answers a specific question: how much of v points along u's own direction? It's computed as the dot product of the two vectors divided by u's own magnitude, proj = (v · u) ⁄ |u|. The dot product v · u already measures how aligned the two vectors are, but its raw value changes if u is simply rescaled to a different length; dividing by |u| removes that dependence, leaving a result that depends only on u's DIRECTION, not on how long the vector u happens to be drawn.
Picture shining a light straight down onto the line that u lies along: v casts a shadow somewhere on that line, and the scalar projection is the signed length of that shadow, measured from the origin. If v points mostly the same way as u, the projection is a sizable positive number; if v points mostly opposite to u, the projection is negative; if v is perfectly perpendicular to u, the projection is exactly zero — v casts no shadow along u's line at all.
This is a genuinely different operation from the already-live Angle Between Two Vectors page elsewhere on this site. Both use the same dot product as a starting ingredient, but one divides by |u| alone to produce a LENGTH along u's direction (this page), while the other divides by |v||u| together and takes an arccosine to produce an ANGLE between the two vectors — same raw ingredient, two different final questions.
- Enter the vector being projected into the Vector v: x and Vector v: y fields.
- Enter the direction it's being projected onto into the Vector u (direction to project onto): x and y fields.
- Read Scalar projection of v onto u: the sheet computes the dot product and divides by u's own magnitude.
- A result of 0 means v is perpendicular to u; a negative result means v points mostly opposite to u's direction.
Worked example — projecting (3, 4) onto the x-axis
Vector v = (3, 4) is projected onto u = (1, 0), the plain x-axis direction. The dot product is v · u = 3×1 + 4×0 = 3, and u's magnitude is |u| = 1, so the projection is proj = 3 ⁄ 1 = 3 — exactly v's own x-component, as it should be, since projecting onto the bare x-axis simply reads off how far along that axis v extends.
Compare projecting v = (5, 0) onto u = (0, 1), straight up the y-axis: the dot product is 5×0 + 0×1 = 0, giving a projection of exactly 0 — v has no component at all along a direction perfectly perpendicular to it, casting no shadow on that line whatsoever.
Questions
What does the scalar projection of one vector onto another mean?
It measures how much of vector v points along vector u's direction, expressed as a single signed length — positive if v leans the same way as u, negative if it leans opposite, and zero if the two are perpendicular. It's the length of the shadow v would cast onto the line u lies along.
What is the formula for scalar projection?
proj = (v · u) ⁄ |u|, the dot product of the two vectors divided by the magnitude of u, the vector being projected onto. Dividing by |u| ensures the result depends only on u's direction, not on how long u itself happens to be.
What does a projection of zero mean?
It means v is exactly perpendicular to u — v has no component at all along u's direction, casting no shadow on that line. This happens whenever the dot product v · u itself equals zero, regardless of either vector's individual length.
How is this different from the angle between two vectors?
Both start from the same dot product, but this page divides by |u| alone to produce a LENGTH along u's direction, while finding the angle between two vectors instead divides by |v||u| together and takes an arccosine to produce an ANGLE — the same ingredient used to answer two different questions.
What is the projection of a vector onto itself?
Its own full magnitude — a vector points 100% along its own direction, so nothing is lost projecting it onto itself. For v = (3, 4), for instance, projecting v onto itself returns exactly |v| = 5.