Compute 3D Vector Norm

3D Geometry

Easy

Problem

Theory

A vector norm is a function that assigns a non-negative length or magnitude to a vector. It measures "how big" the vector is.

The most common norm in 3D geometry is the Euclidean norm (L2 norm), which corresponds to the straight-line distance from the origin.

The Euclidean Norm (L2 Norm)

For a 3D vector \mathbf{v} = (v_x, v_y, v_z):

||\mathbf{v}||_2 = \sqrt{v_x^2 + v_y^2 + v_z^2}

This is the distance from the origin to the point (v_x, v_y, v_z) in 3D space.

When the subscript is omitted, ||\mathbf{v}|| usually means the L2 norm.

Geometric Interpretation

The Euclidean norm represents:

1. Length of the vector:

The straight-line distance from the origin to the vector's tip.

2. Magnitude of a quantity:

Speed (magnitude of velocity), force magnitude, etc.

3. Distance in 3D space:

The distance from point A to point B is ||\mathbf{B} - \mathbf{A}||.

Derivation from Pythagorean Theorem

In 2D, the Pythagorean theorem gives:

||\mathbf{v}||^2 = v_x^2 + v_y^2

In 3D, we apply it twice:

First in the XY-plane: d_{xy}^2 = v_x^2 + v_y^2

Then from XY to Z: ||\mathbf{v}||^2 = d_{xy}^2 + v_z^2 = v_x^2 + v_y^2 + v_z^2

Worked Example 1

Vector: \mathbf{v} = (3, 4, 0)

||\mathbf{v}|| = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{9 + 16 + 0} = \sqrt{25} = 5

This is the classic 3-4-5 right triangle in the XY-plane.

Worked Example 2

Vector: \mathbf{v} = (1, 2, 2)

||\mathbf{v}|| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3

Worked Example 3

Vector: \mathbf{v} = (1, 1, 1)

||\mathbf{v}|| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \approx 1.732

This is the space diagonal of a unit cube.

Properties of Norms

1. Non-negativity:

||\mathbf{v}|| \geq 0

2. Zero only for zero vector:

||\mathbf{v}|| = 0 \iff \mathbf{v} = \mathbf{0}

3. Scalar multiplication:

||c\mathbf{v}|| = |c| \cdot ||\mathbf{v}||

4. Triangle inequality:

||\mathbf{u} + \mathbf{v}|| \leq ||\mathbf{u}|| + ||\mathbf{v}||

The Squared Norm

Often, we use the squared norm to avoid computing square roots:

||\mathbf{v}||^2 = v_x^2 + v_y^2 + v_z^2

Benefits:

• Faster computation (no square root)
• Preserves ordering: ||\mathbf{a}|| < ||\mathbf{b}|| \iff ||\mathbf{a}||^2 < ||\mathbf{b}||^2
• Used in many optimization problems

Relationship to Dot Product

The squared norm equals the dot product with itself:

||\mathbf{v}||^2 = \mathbf{v} \cdot \mathbf{v} = v_x^2 + v_y^2 + v_z^2

Therefore:

||\mathbf{v}|| = \sqrt{\mathbf{v} \cdot \mathbf{v}}

This relationship is fundamental in linear algebra.

Other Common Norms

L1 Norm (Manhattan norm):

||\mathbf{v}||_1 = |v_x| + |v_y| + |v_z|

Distance traveling along axis-aligned paths.

L\infty Norm (Maximum norm):

||\mathbf{v}||_\infty = \max(|v_x|, |v_y|, |v_z|)

The largest component magnitude.

Lp Norm (General):

||\mathbf{v}||_p = \left( |v_x|^p + |v_y|^p + |v_z|^p \right)^{1/p}

L2 is the case p = 2.

Comparing Norms

For the same vector \mathbf{v} = (3, 4, 0):

||\mathbf{v}||_1 = 3 + 4 + 0 = 7

||\mathbf{v}||_2 = \sqrt{9 + 16} = 5

||\mathbf{v}||_\infty = \max(3, 4, 0) = 4

In general: ||\mathbf{v}||_\infty \leq ||\mathbf{v}||_2 \leq ||\mathbf{v}||_1

Distance Between Points

The Euclidean distance between points \mathbf{A} and \mathbf{B}:

d(\mathbf{A}, \mathbf{B}) = ||\mathbf{B} - \mathbf{A}|| = \sqrt{(B_x - A_x)^2 + (B_y - A_y)^2 + (B_z - A_z)^2}

Example: Distance from (1, 2, 3) to (4, 6, 3):

d = \sqrt{(4-1)^2 + (6-2)^2 + (3-3)^2} = \sqrt{9 + 16 + 0} = 5

Unit Vectors

A unit vector has norm equal to 1:

||\hat{\mathbf{v}}|| = 1

To create a unit vector (normalization):

\hat{\mathbf{v}} = \frac{\mathbf{v}}{||\mathbf{v}||}

Unit vectors represent pure direction without magnitude.

Applications in 3D Graphics

Distance calculations:

• Object collision detection
• Level of detail selection
• Culling (discard far objects)

Lighting:

• Light attenuation: I \propto \frac{1}{||\mathbf{r}||^2}
• Normal vectors (must be unit length)

Animation:

• Speed = norm of velocity vector
• Interpolation distances

Applications in Physics

Magnitude of vectors:

• Speed: |v| = ||\mathbf{v}||
• Force: |F| = ||\mathbf{F}||
• Acceleration: |a| = ||\mathbf{a}||

Energy:

• Kinetic energy: KE = \frac{1}{2}m||\mathbf{v}||^2

Work:

• Uses dot product which relates to norms

Applications in Machine Learning

Distance metrics:

• k-NN classification uses Euclidean distance
• Clustering algorithms (k-means)

Regularization:

• L2 regularization (Ridge): \lambda ||\mathbf{w}||^2_2
• L1 regularization (Lasso): \lambda ||\mathbf{w}||_1

Loss functions:

• Mean Squared Error uses squared L2 norm
• Mean Absolute Error uses L1 norm

Numerical Considerations

Overflow:

For very large components, v_x^2 might overflow.

Solution: Use logarithms or scale before squaring.

Underflow:

For very small components, v_x^2 might underflow to zero.

Solution: Use extended precision or relative comparisons.

Catastrophic cancellation:

When components have very different magnitudes, small components may be lost.

Efficient Computation

Avoid redundant square roots:

When comparing distances, use squared norms:

||\mathbf{a}|| < ||\mathbf{b}|| \iff ||\mathbf{a}||^2 < ||\mathbf{b}||^2

SIMD optimization:

Modern CPUs can compute v_x^2 + v_y^2 + v_z^2 in parallel.

Batch computation:

For many vectors, process in batches using matrix operations.

Generalizing to n Dimensions

The formula extends to any dimension:

||\mathbf{v}||_2 = \sqrt{\sum_{i=1}^{n} v_i^2}

In high-dimensional spaces, distances behave differently (curse of dimensionality), but the formula remains the same.

Norm Inequalities

Cauchy-Schwarz inequality:

|\mathbf{u} \cdot \mathbf{v}| \leq ||\mathbf{u}|| \cdot ||\mathbf{v}||

Triangle inequality:

||\mathbf{u} + \mathbf{v}|| \leq ||\mathbf{u}|| + ||\mathbf{v}||

Reverse triangle inequality:

||\mathbf{u} - \mathbf{v}|| \geq | ||\mathbf{u}|| - ||\mathbf{v}|| |

These are fundamental properties used in proofs and algorithms.

Examples

Hints

1. Use np.sum(values ** 2, axis=-1) for both accepted shapes.
2. Convert a zero-dimensional result with float(norms); otherwise return the array.

Requirements

• Compute the sum of squared components along the final axis
• Return a Python float for one vector
• Return a one-dimensional NumPy array for a batch
• Do not loop over batch rows

Constraints

• Input shape is (3,) or (N, 3)
• N <= 100000
• Use NumPy only

Starter Code

import numpy as np

def vector_norm_3d(v: list) -> float | np.ndarray:
    """
    Returns a float or a NumPy array.
    """
    # Write code here
    pass

Test Cases