Vector Projection Calculator

What projecting a vector onto another vector means

Picture a ramp with a crate on it. You push the crate with a force that points partly forward and partly down, but only the forward share of that force moves the crate along the ramp. Vector projection is the math that isolates that share: it answers the question of how much of one vector lies in the direction of another, and it returns the answer both as a signed length and as a full vector sitting along the target direction.

Geometry gives the cleanest definition. Drop a perpendicular from the tip of vector v down to the line that runs through vector u. The foot of that perpendicular marks the projection: a point on the line of u such that the piece you chopped off v is exactly at a right angle to u. Because the leftover piece is perpendicular, nothing of it points along u, which is why the projection is the whole contribution of v in that direction. MIT OpenCourseWare develops this picture in its linear algebra text and shows the projection as the vector on the u line whose length is the length of v multiplied by the cosine of the angle between the two vectors.

The idea splits into two related quantities that people often mix up. The scalar projection, also called the component of v along u, is a single signed number: how long the shadow is, negative if v points against u. The vector projection is the shadow itself, a vector that points along u and has that signed length. This page's calculator returns both, along with the perpendicular piece that remains once the shadow is removed.

The two projection formulas

Both quantities come from the dot product. For a vector v projected onto a vector u, the scalar projection is comp_uv = (u dot v) / |u|, and the vector projection is proj_uv = ((u dot v) / |u|²) u. Dividing by |u|² instead of |u| converts the signed length into the scalar that multiplies u itself, so the result lands on the line of u automatically. OpenStax Calculus Volume 3, section 2.3 lists both forms as its standard definitions.

Work the first example with whole numbers you can check by hand. Take v = (3, 4, 0) and u = (1, 2, 0). The dot product is 1 x 3 + 2 x 4 + 0 x 0 = 11. The squared length of u is 1 + 4 = 5, and the length of u is 2.2361. The scalar projection is 11 / 2.2361 = 4.9193, and the vector projection is (11/5) x (1, 2, 0) = (2.2, 4.4, 0). Subtracting the projection from v leaves (0.8, -0.4, 0), a leftover of length 0.8944 that is perpendicular to u, which you can confirm because its dot product with u is 0.8 - 0.8 = 0.

Three checks catch most arithmetic slips. The projection must be a scalar multiple of u, the leftover v minus proj must have a dot product of zero with u, and the Pythagorean relation |v|² = |proj|² + |leftover|² must hold. Here |v| = 5, and indeed 5 x 5 = 4.9193² + 0.8944² = 24.2 + 0.8 = 25.

Worked example: the share of a force that does work

Projection earns its keep in physics the moment a force and a motion point in different directions. Suppose a cable pulls a sled with a force of F = (50, 20, 0) newtons while the sled moves along a displacement of s = (6, 8, 0) meters. The work done by the force is the dot product F dot s = 50 x 6 + 20 x 8 = 460 joules, but the question a designer usually asks is different: what force acts along the direction of travel?

Project F onto s. The squared length of s is 36 + 64 = 100, so the scalar multiplier is 460 / 100 = 4.6, and the vector projection is 4.6 x (6, 8, 0) = (27.6, 36.8, 0) newtons. The scalar projection, the signed length of that vector, is 460 / 10 = 46 newtons, since |s| = 10 meters. Only that 46 newton share of the pull pushes the sled along its path; the perpendicular leftover, 22.4 and -16.8 components summing to 28 newtons, does no work at all because it is perpendicular to the motion.

Laid out line by line, the arithmetic looks like this, and each row is the input to the next one:

The same projection also hands you the angle for free. The scalar projection equals |F| times the cosine of the angle between the vectors, so cos(angle) = 46 / |F|. The length of F is 50 x 50 + 20 x 20 = 2900 under a square root, about 53.852 newtons, giving cos(angle) = 46 / 53.852 = 0.8542, which is an angle of about 31.3 degrees. If a textbook answer disagrees with that angle, the error is almost always in the dot product, not in the cosine step.

A table of worked projections

The table collects projections the way they show up on homework and on the job, computed with the same two formulas above. Each row is a vector v projected onto a vector u, with the scalar projection and the resulting vector. The last two rows show the special cases worth recognizing on sight: a perpendicular pair projects to zero, and a vector that points against u projects to a negative multiple.

The third row is the exercise from the MIT 18.01 study guide for chapter 11 (2023): B = (1, 3, 1) projected onto A = (2, 1, 2), where the dot product is 7, the squared length of A is 9, and the projection is (7/9) x (2, 1, 2). Reading the rows top to bottom, the pattern is that only the ratio of dot product to squared length matters, which is why the formulas never need the angle directly.

What the sign and the angle tell you

The scalar projection carries a sign, and the sign is information. A positive value means v leans in the same direction as u, a negative value means v leans against u, and zero means the two are perpendicular. That makes the scalar projection the cheapest orthogonality test available: compute one dot product and divide by one length. In quality control and computer graphics the same test decides whether a surface faces a viewer or faces away.

Because comp_uv = |v| cos(angle), the scalar projection is also the leg of a right triangle with |v| as the hypotenuse. Doubling the length of v doubles the projection, while changing the length of u changes nothing about the scalar projection, only about the vector that carries it. A common exam trap rescales u and expects the component to move; it does not, since the component measures v's share, not u's.

The pattern shows up cleanly if you hold the length of v fixed at 5 and sweep the angle while u points along the positive x-axis. The scalar projection is then 5 cos(angle), so the table below is just the cosine curve scaled by five:

Read down the scalar projection column and you have the whole story of the sign: the value falls smoothly from 5 to 0 to -5, crossing zero exactly at the right angle. No separate sign test is needed, because the dot product already carries it.

When you need the perpendicular piece explicitly, subtract: v minus proj_uv is the part of v that u cannot see. Engineers use it to decompose a load into the part a bearing carries axially and the part it sees radially, and navigation uses it to split a wind vector into the share along the runway and the crosswind. The two pieces always satisfy the Pythagorean relation from the second section, which makes them a built in error check on any hand calculation.

Traps that skew a projection

• Dividing by |u| when you need |u|², or the reverse. The scalar projection divides the dot product by the length of u, the vector projection divides it by the squared length. Swapping the two rescales the answer by |u| and turns a length into a multiplier.
• Projecting onto the zero vector. If u = (0, 0, 0) the formulas divide by zero and the projection is undefined. Every direction question needs a nonzero direction, so check u before you compute.
• Reading the vector projection as a length. The vector result is a point in space, not a distance. Report the scalar projection when someone asks how much, and the vector projection when someone asks which way.
• Mixing the direction of projection. comp_uv and comp_vu differ whenever the two lengths differ. Saying "project v onto u" fixes the direction; shorthand like "the projection of the two" does not.
• Unit vectors forgotten at the end. If u is a unit vector the formulas collapse, since |u|² = 1, but a vector of length 2 makes the divisor 4. Normalizing u first is safe only if you remember the scalar projection of the original pair, not the normalized one, when a length is requested.

Neighboring tools for the rest of the vector work

The calculator below runs both formulas at once: enter v and u in two or three dimensions, and it returns the dot product, the scalar projection, the vector projection and the perpendicular leftover. For the pieces this page treats as inputs, the dot product calculator works through the sum of products step by step, the vector magnitude calculator handles the square root of the squared length, and the vector addition calculator covers the componentwise sums that combine vectors back together. When the angle between two vectors is the goal rather than a component, the law of cosines calculator and the distance calculator cover the triangle side of the same geometry.