6.28 min read

Length and Distance

The length (or norm) of a vector $\mathbf{v}$ is the square root of its dot product with itself: $\|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}} = \sqrt{v_1^2 + \cdots + v_m^2}$ . This matches the familiar Pythagorean formula for distance from the origin.

Scaling a vector scales its length proportionally: $\|c\mathbf{v}\| = |c| \cdot \|\mathbf{v}\|$ . A vector with length 1 is called a unit vector. Given any nonzero vector $\mathbf{v}$ , dividing by its length gives a unit vector in the same direction: $\hat{\mathbf{v}} = \mathbf{v} / \|\mathbf{v}\|$ . This is called normalization.

The distance between two vectors $\mathbf{u}$ and $\mathbf{v}$ is the length of their difference: $\text{Dist}(\mathbf{u}, \mathbf{v}) = \|\mathbf{u} - \mathbf{v}\|$ . Think of $\mathbf{u}$ and $\mathbf{v}$ as points, and the vector $\mathbf{u} - \mathbf{v}$ as the arrow from $\mathbf{v}$ to $\mathbf{u}$ .

Formal View

Definition 6.3 — Length (Norm)

For

\mathbf{v} \in \mathbb{R}^m

, its length is

\|\mathbf{v}\| := \sqrt{\mathbf{v} \cdot \mathbf{v}} = \sqrt{\sum_i v_i^2}

. The squared length is

\|\mathbf{v}\|^2 = \mathbf{v}^t \mathbf{v}

. If

\|\mathbf{v}\| = 1

, we call

\mathbf{v}

a unit vector, sometimes written

\hat{\mathbf{v}}

Definition 6.4 — Distance

The distance between

\mathbf{u}, \mathbf{v} \in \mathbb{R}^m

\text{Dist}(\mathbf{u}, \mathbf{v}) := \|\mathbf{u} - \mathbf{v}\| = \sqrt{\sum_i (u_i - v_i)^2}.

Why This Matters

Norms and distances give geometry its measurement capabilities — they let us say what "close" and "far" mean.

K-nearest neighbors classifiers assign a class to a test point based on the Euclidean distances to training examples
Image compression measures reconstruction error by the squared distance $\|\mathbf{x}_{\text{original}} - \mathbf{x}_{\text{compressed}}\|^2$
GPS and sensor fusion use norm minimization to find the most likely position given noisy measurements
The Frobenius norm of a matrix — the length of all entries as one long vector — measures how far a matrix is from zero

Learning Resources

Vector magnitude and normalization

Khan Academy

Computing vector length and normalizing to unit length.

8 min

Norms and distance in linear algebra

MIT OpenCourseWare

Strang introduces norms and distance in the context of orthogonality.

47 min

Quiz

Question 1

What is $\|\mathbf{v}\|$ for $\mathbf{v} = (3, -4)$ ?

Question 2

If $\|\mathbf{v}\| = 4$ , what is $\|3\mathbf{v}\|$ ?

Common Mistakes

Computing $\|\mathbf{u} + \mathbf{v}\|$ as $\|\mathbf{u}\| + \|\mathbf{v}\|$ — the triangle inequality says $\|\mathbf{u} + \mathbf{v}\| \leq \|\mathbf{u}\| + \|\mathbf{v}\|$ , with equality only when they point in the same direction.
Forgetting to take the square root — $\mathbf{v} \cdot \mathbf{v}$ is the squared length, not the length.
Normalizing the zero vector — division by $\|\mathbf{0}\| = 0$ is undefined.