The Pythagorean Identity for Vector Components

Why Perpendicular Components Produce a Sum of Squares

A vector can be split into parts that point in different directions. When those parts are perpendicular, the length of the original vector is linked to the lengths of its components by the same sum-of-squares relationship as the sides of a right triangle.

This identity is useful in both directions: it can calculate the length of a vector from its components, or recover one component length when the other lengths are known. The key condition is perpendicularity. Without it, the same sum-of-squares formula need not hold.

The Pythagorean Identity for Vector Components

Suppose a vector \(\mathbf{v}\) is the sum of two perpendicular vectors, \(\mathbf{a}\) and \(\mathbf{b}\). Then \(\mathbf{a}\) and \(\mathbf{b}\) are its components, and their lengths combine as follows.

Because lengths are nonnegative, the equivalent identity for lengths is \(\|\mathbf{v}\|=\sqrt{\|\mathbf{a}\|^2+\|\mathbf{b}\|^2}\). The squares are essential: lengths themselves do not generally add.

Why the Identity Holds

The proof follows directly from the definition of the norm and the dot product. The squared norm of any vector \(\mathbf{x}\) is \(\mathbf{x}\cdot\mathbf{x}\), and the dot product is distributive over vector addition.

If the components are not perpendicular, the expansion still applies, but the mixed terms generally remain. In that case, the sum of the squared component lengths alone does not give the squared length of their sum.

Parallel and Perpendicular Parts of a Vector

Let \(\mathbf{u}\) be a unit vector, so \(\|\mathbf{u}\|=1\). For any vector \(\mathbf{v}\), its scalar component in the direction \(\mathbf{u}\) is \(\mathbf{v}\cdot\mathbf{u}\). The vector component parallel to \(\mathbf{u}\) is \((\mathbf{v}\cdot\mathbf{u})\mathbf{u}\); subtracting it from \(\mathbf{v}\) gives the perpendicular component.

These components are indeed perpendicular: \(\mathbf{q}\cdot\mathbf{u}=\mathbf{v}\cdot\mathbf{u}-(\mathbf{v}\cdot\mathbf{u})(\mathbf{u}\cdot\mathbf{u})=0\), since \(\mathbf{u}\cdot\mathbf{u}=1\). Therefore \(\mathbf{p}\cdot\mathbf{q}=0\), and the Pythagorean identity gives

The parallel component's length is \(|\mathbf{v}\cdot\mathbf{u}|\), not necessarily \(\mathbf{v}\cdot\mathbf{u}\): the latter is a signed scalar and may be negative. Squaring it gives the same nonnegative contribution to the identity.

Recovering an Unknown Component Length

Rearrange the identity to isolate the unknown squared length. If the total length is \(L\) and one perpendicular component has length \(a\), then the other component has length \(b\) satisfying \(b^2=L^2-a^2\). Since a length cannot be negative, \(b=\sqrt{L^2-a^2}\).

The condition on the input lengths is important. If a proposed total length were \(4\) and a perpendicular component already had length \(5\), the calculation would require the other squared length to be \(4^2-5^2=16-25=-9\). No real length has square \(-9\), so those measurements cannot describe perpendicular components of a vector of length \(4\).

What the Squared Identity Does—and Does Not—Tell You

When solving for a length, the nonnegative square root is the correct choice. When solving for a signed scalar component, the squared equation by itself may leave two possibilities. The sign must come from other information, such as whether the vector points with or against the chosen unit direction.

There is also a useful feasibility check. If \(L\) is the total length and \(a\) is the length of one component, then a perpendicular decomposition requires \(L^2-a^2\geq 0\). If this condition fails, the measurements are inconsistent. If it holds, the other component's length is the nonnegative square root of that difference.

A Common Mistake: Adding Lengths Instead of Their Squares

A frequent error is to treat the perpendicular component lengths as if they added directly. For example, if a vector has length \(13\) and its parallel component has length \(5\), the incorrect calculation would be \(13=5+b\), giving \(b=8\). But checking that answer gives \(5^2+8^2=25+64=89\), not \(13^2=169\).

The correction is to apply the identity to squared lengths: \(13^2=5^2+b^2\), so \(b^2=169-25=144\), and \(b=12\). Checking gives \(5^2+12^2=25+144=169=13^2\). The mistake happened because perpendicular lengths form a right-triangle relationship: the squares, not the lengths, satisfy the sum.