Orthogonal Vectors and Subspaces

Orthogonality from the dot product

Two vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ are orthogonal (written $\mathbf{u} \perp \mathbf{v}$) if their dot product is zero: $\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n = 0$

This is equivalent to the geometric statement "the angle between them is $90°$" — because $\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \, \|\mathbf{v}\| \, \cos \theta$

and $\cos 90° = 0$.

Step-by-step

$\mathbf{u} = (1, 2, -1), \quad \mathbf{v} = (3, 0, 3)$.

Step 1 — Compute the dot product. $\mathbf{u} \cdot \mathbf{v} = (1)(3) + (2)(0) + (-1)(3)$

Step 2 — Evaluate. $\mathbf{u} \cdot \mathbf{v} = 3 + 0 - 3 = \boxed{0}$

So $\mathbf{u} \perp \mathbf{v}$ ✓

Angle: $\cos \theta = 0 / (\|\mathbf{u}\| \|\mathbf{v}\|) = 0$, so $\theta = 90°$.

Norms and normalization

The norm (length) of a vector is $\|\mathbf{u}\| = \sqrt{\mathbf{u} \cdot \mathbf{u}}$.

For our $\mathbf{u}$: $\|\mathbf{u}\| = \sqrt{1 + 4 + 1} = \sqrt{6}$. For $\mathbf{v}$: $\|\mathbf{v}\| = \sqrt{9 + 0 + 9} = \sqrt{18} = 3\sqrt{2}$.

A unit vector $\hat{\mathbf{u}} = \mathbf{u}/\|\mathbf{u}\|$ has norm 1. Two unit vectors are orthonormal iff they are orthogonal.

Properties of the dot product

• Commutative: $\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}$.
• Distributive: $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}$.
• Scalar: $(c \mathbf{u}) \cdot \mathbf{v} = c (\mathbf{u} \cdot \mathbf{v})$.
• Positive-definite: $\mathbf{u} \cdot \mathbf{u} \ge 0$, with equality iff $\mathbf{u} = \mathbf{0}$.

Orthogonal subspaces

Two subspaces $U, W$ are orthogonal if every vector in $U$ is orthogonal to every vector in $W$. They need not span the whole space; they just can't share any direction.

The orthogonal complement of $U$, written $U^\perp$, is the set of all vectors orthogonal to every vector in $U$. It satisfies: $\dim U + \dim U^\perp = \dim (\text{ambient space})$

Important identities for any $m \times n$ matrix $A$:

• $\operatorname{Row}(A)^\perp = \operatorname{Null}(A)$
• $\operatorname{Col}(A)^\perp = \operatorname{Null}(A^T)$

These are the four fundamental subspaces.

Why orthogonality matters

• Pythagorean theorem: $\mathbf{u} \perp \mathbf{v} \implies \|\mathbf{u} + \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2$.
• Projection: projecting onto a subspace uses orthogonal decomposition.
• Gram-Schmidt: builds an orthogonal basis from any set of linearly independent vectors.
• Fourier analysis: the sines and cosines $\{1, \cos nx, \sin nx\}$ are mutually orthogonal under the integral inner product.

Common mistakes

• Calling vectors orthogonal when one is zero. By convention, $\mathbf{0}$ is orthogonal to every vector (dot product is zero).
• Confusing orthogonality with parallelism. Orthogonal: dot product 0. Parallel: cross product 0 (in $\mathbb{R}^3$) or vectors are scalar multiples.
• Forgetting to square-root the norm. $\|\mathbf{u}\|^2 = \mathbf{u} \cdot \mathbf{u}$; the length itself takes a square root.

Try it in the visualization

Drag two vectors in 3D. The angle display updates; the two vectors are colored to show orthogonality (green) vs. acute/obtuse angles (yellow/red). A small right-angle square appears at the origin when the dot product is zero.