多维度下的向量 [Math 241] - Section 12.1

# The Key Spaces —— 空间维度

$\mathbb{R}$: real numbers - 实数
e.x: 2, 0, -0.5, $\pi$, ...

$\mathbb{R} ^ 2$: Two-dimensional plane/space - 二维平面

point in 2-dim space is pair of real numbers, such as (x, y)
二维平面中的点由一对实数组成

$\mathbb{R} ^ 3$: Three-dimensional plane/space - 三维平面

point in 3-dim space is triple of real numbers, such as (x, y, z)

$\mathbb{R} ^ n$: n-dimensional plane/space - n 维平面

n-tuples of real numbers

# Vectors —— 向量

Motivation: Want to describe mathmatrically quantities, such as force, wind.

# geometric point of view

determined by:

• direction
• length / magnitude

e.x.

works only in $\mathbb{R} ^ 2$ and $\mathbb{R} ^ 3$
一般只能用来描述二维与三维，因为再高纬度我们也画不出来了。

# algebraic point of view

n-tuples of real numbers

Vectors in $\mathbb{R} ^ n$:
$\vec {a} = <a_1, a_2, ..., a_n>$
$a_1, a_2, ..., a_n$ are all real numbers

works only in any dimension

e.x.

$\vec {a} = <a_1, a_2>$
$\vec {}$: arrow notaction
< > : angle brackets
$a_1$ : x-comprment
$a_2$ : y-comprment