Vector Projection Calculator
This vector projection calculator works out how much of one vector lies along another, and keeps the two answers firmly apart — because they are not the same kind of thing.
Type <3, 4> . <1, 1> and you get the arrow (7/2, 7/2) and the length 7√2/2. One of those is a vector. The other is a number.
Think of a Shadow
Stand an arrow up against a wall and shine a light straight down the wall. The patch of wall the arrow covers is its projection.
Two sensible questions follow, and they have different answers. How far along the wall does the shadow reach? — a distance. What does the shadow look like as an arrow lying on the wall? — a vector.
Both are called the projection, which is exactly why students mix them up. This page gives both and labels them so you cannot copy the wrong one by accident.
Which Onto Which
Swap the two vectors and the answer changes completely. It is not a subtlety, it is a different question.
⟨3, 4⟩ onto ⟨1, 0⟩ gives (3, 0). ⟨1, 0⟩ onto ⟨3, 4⟩ gives (3/25, 4/25). Different direction, different length, and only one of them is what your question asked for.
The wall is the second vector. Whatever is named second is what everything gets projected onto. This page reads that back to you in full, in words, before showing any numbers.
One Ratio Does All the Work
Everything comes out of a single number: (a · b) ÷ (b · b).
That ratio says how many copies of the second vector make up the shadow. For ⟨3, 4⟩ onto ⟨1, 1⟩ it is 7/2, so the shadow is three and a half copies of ⟨1, 1⟩ — which is (7/2, 7/2).
Notice what is not there. No square root. Dividing by b · b means dividing by a length squared, and the roots cancel, so the vector answer is always an exact fraction. That is worth knowing because the other answer is not.
Why the Length Carries a Root
The scalar projection divides by |b| rather than by b · b. A length is usually a square root, so one usually survives.
For ⟨3, 4⟩ onto ⟨1, 1⟩ it is 7 ÷ √2, which properly written is 7√2/2, about 4.9497. Rounding it to 4.95 and carrying that forward is how a clean question turns into an untidy answer three lines later.
| Vector projection | Scalar projection | |
|---|---|---|
| What it is | an arrow | a length |
| Divide by | b · b | |b| |
| Can it be negative? | no, it reverses instead | yes |
| Roots involved? | never | usually |
A Negative Answer Is Not a Mistake
Project ⟨3, 4⟩ onto ⟨−1, 0⟩ and the scalar projection comes to −3.
The minus sign is doing real work. It says the first vector leans away from the second, so the shadow falls on the far side of where the wall points. Strip the sign off and you have thrown that away — and in a physics question it is the difference between a force helping and a force resisting.
The vector projection for that same pair is (3, 0), with no minus in sight. Nothing has gone wrong: the arrow points where it points, and the sign lives in the scalar version instead.
What Is Left Over Proves the Answer
Take the projection away from your original vector. What remains is called the rejection, and it always sits at right angles to the second vector.
That gives the strongest check available on this whole page, and it costs two lines. Add the projection and the rejection: you must get your original vector back, component for component. Then dot the rejection with the second vector: it must be exactly zero.
For ⟨3, 4⟩ onto ⟨1, 1⟩ the pieces are (7/2, 7/2) and (−1/2, 1/2). They add to (3, 4). The second dotted with ⟨1, 1⟩ gives −1/2 + 1/2 = 0. Both hold, so the answer is right — and this page runs both in whole-number fractions before printing anything at all.
Where the Shadow Actually Gets Used
Almost never as an exercise in its own right.
Push a sledge with a rope held at an angle and only part of your pull moves it forwards; the rest lifts uselessly against the ground. That forward part is the projection of the force onto the direction of travel, and the work you do is that projection times the distance. Change the angle of the rope and you change the projection, which is why a low rope beats a high one and why the answer to "how hard should I pull" is really a question about direction.
The same shape of calculation turns up wherever something has to be split into a useful part and a wasted one. A satellite dish points at a signal and receives the projection of it. A machine-learning model compares two documents by how much one lies along the other. A physics engine bounces a ball off a wall by splitting its velocity into the part along the wall, which survives, and the part into it, which reverses — and that second piece is precisely the rejection this page prints.
None of that needs a different formula. It is the same ratio every time.
How to Use This Vector Projection Calculator
First the vector being projected, then the one it is projected onto, each bracketed: <3, 4> . <1, 0>. Two components or three. Fractions in, fractions out.
The band gives the arrow, the length with its sign, and the leftover. Show the working opens the ratio it all rests on, both answers derived separately, and the two checks written out. The number that starts it all is a dot product, the direction alone would be a unit vector, and if you want the angle rather than the shadow that is the angle between two vectors.
Vector Projection Calculator FAQ
What is the difference between scalar and vector projection?
How do you project one vector onto another?
Does the order matter in a vector projection?
Can a scalar projection be negative?
What is vector rejection?
Why is the vector projection a fraction but the scalar one a root?
What does a zero projection mean?
Can you project onto the zero vector?
Name the wall second, keep the sign, and add the pieces back to be sure.