## Vector Norm

Based on Pythagorean Theorem, the vector from the origin to the point (3, 4) in 2D Euclidean plane has length of $$\sqrt{3^{2}+4^{2}}=\sqrt{25}=5$$ and the vector from the origin to the point $$(a,b)$$ has length $$\sqrt{a^{2}+b^{2}}$$. The length of a vector with two elements is the square root of the sum of each element squared.

The magnitude of a vector is sometimes called the length of a vector, or norm of a vector. Basically, norm of a vector is a measure of distance, symbolized by double vertical bar $$\left \| \mathbf{a} \right \|$$
The magnitude of a vector can be extended to $$n$$ dimensions. A vector a with $$n$$ elements has length
$$\left \| \mathbf{a} \right \| = \sqrt{a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}}$$

The vector length is called Euclidean length or Euclidean norm. Mathematician often used term norm instead of length. Vector norm is defined as any function that associated a scalar with a vector and obeys the three rules below

1. Norm of a vector is always positive or zero $$\left \| \mathbf{a} \right \| \geqslant 0$$. The norm of a vector is zero if and only if the vector is a zero vector $$\mathbf{a} = \mathbf{0}$$.
2. A scalar multiple to a norm is equal to the product of the absolute value of the scalar and the norm$$\left \| k\mathbf{a} \right \|=\left | k \right |\left \| \mathbf{a} \right \|$$.
3. Norm of a vector obeys triangular inequality that the norm of a sum of two vectors is less than or equal to the sum of the norms $$\left \| \mathbf{a} + \mathbf{b}\right \| \leqslant \left \| \mathbf{a} \right \| + \left \| \mathbf{b} \right \|$$.

There are many common norms:

• 1-norm is defined by the sum of absolute value of the vector elements
$$\left \| \mathbf{a} \right \|_{1} = \left | a_{1} \right | + \left | a_{2} \right | + ... + \left | a_{n} \right |$$ .
• 2-norm is the most often used vector norm, sometimes called Euclidean norm. When the subscript index of the vector norm is not specified, you may think that it is a Euclidean norm $$\left \| \mathbf{a} \right \| = \left \| \mathbf{a} \right \|_{2} = \sqrt{a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}}$$ .
• p-norm is sometimes called Minskowski norm is defined as $$\left \| \mathbf{a} \right \|_{p} = \sqrt[p]{a_{1}^{p}+a_{2}^{p}+...+a_{n}^{p}}$$ . p-norm is generalized norm with a parameter $$p$$ .
• max-norm is also called Chebyshev norm is the largest absolute element in the vector $$\left \| \mathbf{a} \right \|_{\infty} = max[\left | a_{1} \right | , \left | a_{2} \right | , ... , \left | a_{n} \right | ]$$

Use the interactive program below to experiment with your own vector input. The program will give you the norm of vector for p=1, 2, 3 and max. The vector input will be redrawn to give you feedback on what you type. Click Random Example button to generate random vector.

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## Properties

Some important properties of vector norm are

• Square of Euclidean norm is equal to the sum of square $$\left \| \mathbf{a} \right \|^{2} = a_{1}^{2} + a_{2}^{2} + ... + a_{n}^{2}$$.
• Pythagorean Theorem is hold if and only if the two vectors are orthogonal $$\left \| \mathbf{a}-\mathbf{b} \right \|^{2} = \left \| \mathbf{a} \right \|^{2}+\left \| \mathbf{b} \right \|^{2}, \mathbf{a} \perp \mathbf{b}$$.
• The law of cosine $$\left \| \mathbf{a}-\mathbf{b} \right \|^{2} = \left \| \mathbf{a} \right \|^{2}+\left \| \mathbf{b} \right \|^{2}-2\left \| \mathbf{a} \right \|\left \| \mathbf{b} \right \|cos\phi$$
• Norm of the dot product of two vectors is equal to the product of their norms $$\left \| \mathbf{a}^{T}\mathbf{b} \right \| = \left \| \mathbf{a} \right \|\left \| \mathbf{b} \right \|$$.
• Relationship to vector inner products
• Square of Euclidean norm of a vector is equal to the inner product to itself $$\left \| \mathbf{a} \right \|^{2} = \mathbf{a}^{T}\mathbf{a}$$
• $$\mathbf{a}^{T}\mathbf{b}=\left \| \mathbf{a} \right \|\left \| \mathbf{b} \right \|cos\phi$$ where, $$\phi$$ is the angle between the two vectors
• $$\mathbf{a}^{T}\mathbf{b} = \frac{1}{4} (\left \| \mathbf{a}+\mathbf{b} \right \|^{2}-\left \| \mathbf{a}-\mathbf{b} \right \|^{2})$$
• Norm of addition or subtraction follow the law of cosine $$\left \| \mathbf{a} \pm \mathbf{b} \right \|^{2} = \left \| \mathbf{a} \right \|^{2} + \left \| \mathbf{b} \right \|^{2} \pm 2 \mathbf{a}^{T} \mathbf{b}$$
• Addition of two square of norm of vectors follow parallelogram law $$\left \| \mathbf{a} \right \|^{2} + \left \| \mathbf{b} \right \|^{2} =\frac{1}{2}(\left \| \mathbf{a} + \mathbf{b} \right \|^{2} + \left \| \mathbf{a} - \mathbf{b} \right \|^{2})$$
• p-norm is greater than the max-norm but less than $$n^{\frac{1}{p}}$$ times the max-norm, that is $$1\leqslant \frac{\left \| \mathbf{a} \right \|_{p}}{\left \| \mathbf{a} \right \|_{\infty }}\leq n^{\frac{1}{p}}$$ .
• The norm ratio satisfies the inequality $$1\leqslant \frac{\left \| \mathbf{a} \right \|_{p}}{\left \| \mathbf{a} \right \|_{q }}\leq n^{\frac{q-p}{pq}}$$ . As $$q$$ tends to infinity, the $$\frac{q-p}{pq}$$ approaches $$\frac{1}{p}$$ and $$\left \| \mathbf{a} \right \|_{p}$$ approaches $$\left \| \mathbf{a} \right \|_{\infty}$$.
• Cauchy-Schwartz inequality stated that the absolute value of vector dot product is always less than or equal to the product of their norms $$\left | \mathbf{a}^{T}\mathbf{b} \right |\leq \left \| \mathbf{a} \right \|\left \| \mathbf{b} \right \|$$ . The equality $$\left | \mathbf{a}^{T}\mathbf{b} \right | = \left \| \mathbf{a} \right \|\left \| \mathbf{b} \right \|$$ holds if and only if the vectors are linearly dependent.
• Relationship of norm of cross product and dot product is $$\left \| \mathbf{a} \times \mathbf{b} \right \|^{2} + \left \| \mathbf{a}\cdot \mathbf{b} \right \|^{2}=\left \| \mathbf{a} \right \|^{2}\left \| \mathbf{b} \right \|^{2}$$ .