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vectors.md (1041B)

      1 # Vectors
      2 
      3 A vector is an object that has both **magnitude** (length) and **direction** and
      4 is often represented as an ordered list of numbers, like components along
      5 coordinate axes. In $\mathbb{R}^n$, a vector is typically written as a column or
      6 row of $n$ real numbers and can be added to other vectors or scaled by real
      7 numbers.
      8 
      9 1. Basic vector notation (inline):
     10 
     11 A vector in 2D can be written as $\vec{v} = (v_1, v_2)$.
     12 
     13 2. Column vector (display):
     14 
     15 A column vector in 3D:
     16 
     17 $$
     18 \vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}
     19 $$
     20 
     21 3. Vector in $\mathbb{R}^n$:
     22 
     23 In general, a vector in $\mathbb{R}^n$ is
     24 
     25 $$
     26 \vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \dots \\ v_n \end{bmatrix}.
     27 $$
     28 
     29 4. Vector addition and scalar multiplication:
     30 
     31 If $\vec{u} = (u_1, u_2)$ and $\vec{v} = (v_1, v_2)$, then
     32 
     33 $$
     34 \vec{u} + \vec{v} = (u_1 + v_1,\; u_2 + v_2)
     35 $$
     36 
     37 and for a scalar $a$,
     38 
     39 $$
     40 a\vec{v} = (av_1,\; av_2).
     41 $$
     42 
     43 5. Magnitude (length) of a vector:
     44 
     45 The length of $\vec{v} = (v_1, v_2, v_3)$ is
     46 
     47 $$
     48 \|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}.
     49 $$