Base of multivariate vectors

2025-06-26

What is a vector? For natural number $n$ will call a n-variate vector list of n ordered real numbers $v = \begin{pmatrix}v_1&v_2&v_3&\ldots&v_n\end{pmatrix}$. Well the $n$ number is a vector dimension: $\dim{v} = n$. For $n$ number the set of all possible vectors will call $R^n$. The regular number we call “scalar”. For example, $5 \ - \ scalar$, but $\begin{pmatrix}1&3\end{pmatrix} \ - \ vector$.

The basic operations of vectors

1. Sum of vectors is a sum of corresponding coordinates: $$ v + w = \begin{pmatrix}v_1 \\ v_2 \\ \ldots \\ v_n \end{pmatrix} + \begin{pmatrix}w_1 \\ w_2 \\ \ldots \\ w_n \end{pmatrix} = \begin{pmatrix}v_1 + w_1 \\ v_2 + w_2 \\ \ldots \\ v_n + w_n \end{pmatrix} $$

We can add vectors only they have same dimension.

1. Product of vector by scalar is a product off each coordinates by a scalar: $$ \lambda v = \lambda \cdot \begin{pmatrix}v_1 \\ v_2 \\ \ldots \\ v_n \end{pmatrix} = \begin{pmatrix}\lambda v_1 \\ \lambda v_2 \\ \ldots \\ \lambda v_n \end{pmatrix} $$
2. Difference of vectors is a sum of vectors, but one of vector product by scalar $-1$.
3. Scalar product of vectors: $$ x \cdot y = \langle x;y\rangle = x_1 y_1 + x_2 y_2 + \ldots + x_n y_n = \sum \limits_{i = 1}^{n}x_i y_i $$ Properties of scalar vectors product:

• Commutativity: $\langle v; w\rangle = \langle w; v\rangle$
• Distributivity over addition: $\langle x; (v + w)\rangle = \langle x; v\rangle + \langle x; w \rangle$
• Compatibility with scalar multiplication: $\langle v; \lambda w\rangle = \lambda \langle v;w\rangle$
• If scalar product equal zero we call this two vectors orthogonal or perpendicular

Geometric meaning

In geometry vector is a segment having a direction.

The length of vector we calculate with Pythagoras’s theorem: $||v|| = \sqrt{\sum\limits_{i = 1}^{n}{x_i}}$. If length of vector equal one we call it a unit vector. If $v$ not a unit vector we can change it multiplying by $\lambda = \frac{1}{|v|}$. We can calculate distance and angle between two vectors. Distance is a length of difference two vectors: $||d|| = ||v - w|| = \sqrt{\sum\limits_{i = 1}^{n}{(v_i - w_i)^2}}$. Angle between two vectors follows from Law of Cosines:

Often be helpful: $||u|| = \sqrt{\langle u;u\rangle}$. Next operation is projection vector $u$ to vector $w$:

Another property of vectors: module of scalar product does not exceed product of their length — the Cauchy–Schwarz inequality:

The Triangle inequality: $||v + w|| \leqslant ||v|| + ||w||$, so length of sum of vectors less than or equal to sum of vectors lengths. If vectors is orthogonal, work the Pythagoras’s theorem: $||a + b||^2 = ||a||^2 + ||b||^2$.