Compute the projection of one vector onto another. Enter vectors; get formula, scalar component, projection vector, and perpendicular component instantly.
3, 4 (2D) or 1, 0, 3 (3D). Use commas to separate components.
u and v must have the same number of components.
v·v in the denominator.
Spaces are OK. Decimals allowed.
projv(u) = ((u·v)/(v·v)) v
3,4 or 1,0,3).projv(u) = ((u·v)/(v·v)) v.(u·v)/||v||.
Helping notes: If v = 0 or dimensions mismatch, projection is undefined. Check inputs and try again.
An Orthogonal Projection Calculator finds the closest point in a line, plane, or higher–dimensional subspace to a given vector. Geometrically, it drops a perpendicular from the vector to the subspace, splitting the vector into a “parallel” component that lies inside the subspace and a “perpendicular” component orthogonal to it. Algebraically, the calculator evaluates projections onto a single direction, onto a plane defined by an orthonormal basis, or onto the column space of a full–rank matrix. It also returns the projection matrix, the orthogonal residual, and the distance (norm of the residual), presenting all steps with readable LaTeX.
For a nonzero vector \(\mathbf{u}\), the projection of \(\mathbf{v}\) onto the line spanned by \(\mathbf{u}\) is
\[ \operatorname{proj}_{\mathbf{u}}(\mathbf{v})=\frac{\mathbf{u}^\top \mathbf{v}}{\mathbf{u}^\top \mathbf{u}}\;\mathbf{u}. \]
If \(\mathbf{u}\) is unit, the projection matrix is \(P=\mathbf{u}\mathbf{u}^\top\). For an orthonormal basis \(U=[\mathbf{u}_1\,\cdots\,\mathbf{u}_k]\), the projection onto \(\mathcal{S}=\operatorname{span}(U)\) is
\[ P = U U^\top,\qquad \operatorname{proj}_{\mathcal{S}}(\mathbf{v}) = P\,\mathbf{v},\qquad \mathbf{r}=\mathbf{v}-P\mathbf{v}\ \perp\ \mathcal{S}. \]
For a general full–rank matrix \(A\in\mathbb{R}^{m\times n}\) (columns span the target subspace), the projector is
\[ P = A\,(A^\top A)^{-1}A^\top,\qquad \operatorname{proj}_{\operatorname{Col}(A)}(\mathbf{v})=P\,\mathbf{v},\qquad \|\mathbf{v}-P\mathbf{v}\|=\min_{\mathbf{w}\in\operatorname{Col}(A)}\|\mathbf{v}-\mathbf{w}\|. \]
A plane with unit normal \(\mathbf{n}\) uses \(\operatorname{proj}_{\text{plane}}(\mathbf{v})=\mathbf{v}-(\mathbf{n}^\top\mathbf{v})\mathbf{n}\). The calculator also highlights projector properties \(P^\top=P\) and \(P^2=P\), and it can report the least–squares solution \(\hat{\mathbf{x}}\) of \(A\mathbf{x}\approx\mathbf{b}\) via normal equations \(A^\top A\hat{\mathbf{x}}=A^\top\mathbf{b}\) with \(\operatorname{proj}(\mathbf{b})=A\hat{\mathbf{x}}\).
Line (nonunit \(\mathbf{u}\)): \[ \operatorname{proj}_{\mathbf{u}}(\mathbf{v})=\frac{\mathbf{u}^\top \mathbf{v}}{\mathbf{u}^\top \mathbf{u}}\mathbf{u},\quad P=\frac{\mathbf{u}\mathbf{u}^\top}{\mathbf{u}^\top\mathbf{u}}. \]
Orthonormal basis \(U\): \[ P=UU^\top,\quad \mathbf{v}=P\mathbf{v}+(I-P)\mathbf{v},\ (I-P)\mathbf{v}\perp \operatorname{span}(U). \]
General subspace (full column rank \(A\)): \[ P=A(A^\top A)^{-1}A^\top. \]
Plane with unit normal \(\mathbf{n}\): \[ \operatorname{proj}_{\text{plane}}(\mathbf{v})=\mathbf{v}-(\mathbf{n}^\top\mathbf{v})\mathbf{n}. \]
Least squares link: \[ A^\top A\hat{\mathbf{x}}=A^\top\mathbf{b},\quad \operatorname{proj}_{\operatorname{Col}(A)}(\mathbf{b})=A\hat{\mathbf{x}}. \]
\(\mathbf{u}=(1,2,2)\), \(\mathbf{v}=(2,1,0)\). \(\mathbf{u}^\top\mathbf{v}=4\), \(\mathbf{u}^\top\mathbf{u}=9\). \(\operatorname{proj}_{\mathbf{u}}(\mathbf{v})=\tfrac{4}{9}(1,2,2)=(\tfrac{4}{9},\tfrac{8}{9},\tfrac{8}{9})\). Residual \((\tfrac{14}{9},\tfrac{1}{9},-\tfrac{8}{9})\) with distance \(\sqrt{261}/9\approx1.795\).
Orthonormal basis \(U=[\mathbf{e}_1,\mathbf{e}_2]\Rightarrow P=\operatorname{diag}(1,1,0)\). For \(\mathbf{v}=(1,2,3)\), \(P\mathbf{v}=(1,2,0)\), residual \((0,0,3)\), distance \(3\).
\(A=\begin{bmatrix}1&0\\1&1\\1&2\end{bmatrix}\), \(\mathbf{b}=\begin{bmatrix}1\\2\\2\end{bmatrix}\). \(A^\top A=\begin{bmatrix}3&3\\3&5\end{bmatrix}\), \(A^\top\mathbf{b}=\begin{bmatrix}5\\6\end{bmatrix}\). \(\hat{\mathbf{x}}=(A^\top A)^{-1}A^\top\mathbf{b}=\big(\tfrac{7}{6},\tfrac{1}{2}\big)\). Projection \(A\hat{\mathbf{x}}=\left(\tfrac{7}{6},\tfrac{5}{3},\tfrac{13}{6}\right)\).
Onto a vector uses a single direction; onto a subspace uses many directions, summarized by a projection matrix \(P\).
Normalize \(\mathbf{n}\) (or divide by \(\mathbf{n}^\top\mathbf{n}\)) and use \(\mathbf{v}-(\mathbf{n}^\top\mathbf{v})\mathbf{n}\).
Because \(P\mathbf{v}\) minimizes \(\|\mathbf{v}-\mathbf{w}\|\) over the subspace; the residual is orthogonal, giving the shortest distance.
Use QR/SVD or the pseudoinverse \(A^+\); then \(P=A A^+\) still projects onto \(\operatorname{Col}(A)\).
Yes—replace transposes by conjugate transposes: \(P=A(A^*A)^{-1}A^*\).
Symmetric and idempotent: \(P^\top=P\) and \(P^2=P\). Eigenvalues are \(0\) or \(1\).