Concepts and Properties of Vectors

Definition and Representation of Vectors

中文：向量是有方向和大小的量，用有向线段表示。记作 $\vec{a}$，或用黑体 $\mathbf{a}$。起点为 $O$、终点为 $A$ 的向量记 $\overrightarrow{OA}$。 自由向量：只考虑大小和方向，与位置无关；位置向量：以原点为起点。 在平面/空间坐标系中，向量可表示为坐标形式：$\vec{a} = (x, y)$ 或 $\vec{a} = (x, y, z)$。

English：A vector is a quantity with both magnitude and direction, represented by a directed line segment, denoted $\vec{a}$ or $\mathbf{a}$. The vector from origin $O$ to point $A$ is $\overrightarrow{OA}$. Free vectors depend only on magnitude and direction (position-independent); position vectors start at the origin. In coordinates: $\vec{a} = (x, y)$ or $\vec{a} = (x, y, z)$.

Magnitude and Direction

向量的模与方向

中文：向量的模（长度）$|\vec{a}|$ 或 $\|\vec{a}\|$，非负实数。零向量 $\vec{0}$ 模为0，无方向。单位向量：模为1的向量，记 $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$​。方向：与 $x$-轴夹角或用方向余弦表示。

English：The magnitude (length) is $|\vec{a}|$ or $\|\vec{a}\|$, a non-negative real number. The zero vector $\vec{0}$has magnitude 0 and no direction. A unit vector has magnitude 1: $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$. Direction can be given by the angle with the x-axis or direction cosines.

Equal and Opposite Vectors

相等向量与相反向量

中文：两个向量相等 $\vec{a} = \vec{b}$ 当且仅当模相等且方向相同（平行且同向）。相反向量 $\vec{a}$ 与 $-\vec{a}$ 模相等、方向相反。 English：$\vec{a} = \vec{b}$ if they have equal magnitude and the same direction (parallel and same sense). The opposite vector is $-\vec{a}$ (same magnitude, opposite direction).

Vector Addition and Subtraction

向量的加法与减法

中文：三角形法则或平行四边形法则：$\vec{a} + \vec{b}$ 从 $\vec{a}$终点作 $\vec{b}$，或平行四边形对角线。 减法：$\vec{a} – \vec{b} = \vec{a} + (-\vec{b})$。 坐标运算：若 $\vec{a} = (x_1, y_1)$，$\vec{b} = (x_2, y_2)$，则 $\vec{a} + \vec{b} = (x_1+x_2, y_1+y_2)$。

English：Triangle law or parallelogram law: $\vec{a} + \vec{b}$ is the diagonal. Subtraction: $\vec{a} – \vec{b} = \vec{a} + (-\vec{b})$. In coordinates: component-wise addition.

Scalar Multiplication

数乘向量

中文：$k\vec{a}$（$k$ 为实数）：模变为 $|k| \cdot |\vec{a}|$，$k > 0$同向，$k < 0$反向，$k=0$为零向量。 线性组合：$m\vec{a} + n\vec{b}$。

English：$k\vec{a}$ scales magnitude by $|k|$ and keeps/reverses direction depending on the sign of $k$. Linear combination: $m\vec{a} + n\vec{b}$.

The first key operation is scalar multiplication, multiplying a scalar and a vector. If k is a scalar and $\vec{v}$ is a vector, their product k $\vec{v}$ is defined as follows:

• If k>0, then k $\vec{v}$ is the vector pointing in the same direction as  $\vec{v}$ that’s k times as long as  $\vec{v}$.
• If k=0, then k $\vec{v}$ is  $\vec{0}$.
• If k<0, then k $\vec{v}$ is the vector pointing in the opposite direction from  $\vec{v}$ that’s |k| times as long as  $\vec{v}$.

For example, if →v is the vector shown at left below, here’s how you’d picture 2 $\vec{v}$ and −0.6 $\vec{v}$:

Note that the picture above could be happening in  $\mathbb{R}^2$ or $\mathbb{R}^3$

Fundamental Properties

向量基本性质

交换律：$\vec{a} + \vec{b} = \vec{b} + \vec{a}$

结合律：$(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$

零向量：$\vec{a} + \vec{0} = \vec{a}$

逆向量：$\vec{a} + (-\vec{a}) = \vec{0}$

数乘分配律：$k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$ ，$(k+m)\vec{a} = k\vec{a} + m\vec{a}$

向量共线定理：$\vec{a} \parallel \vec{b}$（非零）$\iff$存在唯一实数 $k$使 $\vec{a} = k\vec{b}$。

English：Commutative, associative, identity (zero vector), inverse laws for addition; distributive laws for scalar multiplication. Collinear vectors: $\vec{a} \parallel \vec{b}$(non-zero) if $\vec{a} = k\vec{b}$ for some unique real $k$.