Scalars and Vectors · 标量和矢量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| scalar/ˈskeɪlə/ | 标量 | biāo liàng |
| vector/ˈvektə/ | 矢量 | shǐ liàng |
| magnitude/ˈmæɡnɪtjuːd/ | 大小 | dà xiǎo |
| components/kəmˈpəʊnənts/ | 分量 | fèn liàng |
"Go 5 km" -- but which way?
- Tell a friend to "walk 5 km" and they could end up anywhere on a circle around you.
- Say "walk 5 km north" and there is exactly one destination.
- Some quantities need only a size; others need a direction too.
- Sorting them out is the first step in all of mechanics.
"走 5 公里"——但往哪个方向?
- 让朋友"走 5 公里",他可能落在你周围圆圈上的任何地方。
- 说"往北走 5 公里",就只有一个终点。
- 有些量只需要一个大小;另一些还需要方向。
- 把它们分清楚,是一切力学的第一步。
Scalars and vectors
- A scalar 标量 has only a size (magnitude) -- like temperature or time.
- A vector 矢量 has a size and a direction -- like a push or a velocity.
- Getting the type right changes how you add them.
标量与矢量
- 标量只有大小(数值)——如温度或时间。
- 矢量既有大小又有方向——如推力或速度。
- 弄对类型,会改变你相加它们的方式。
A quantity with both a size and a direction is called a . · 同时具有大小和方向的量称为。
A vector has magnitude and direction; a scalar has only magnitude. · 矢量有大小和方向；标量只有大小。
Which is which
- Scalars: distance, speed, mass, energy, time.
- Vectors: displacement, velocity, acceleration, force, momentum.
- The vector twin of a scalar carries the extra "which way" information.
哪个是哪个
- 标量:距离、速率、质量、能量、时间。
- 矢量:位移、速度、加速度、力、动量。
- 一个标量的矢量"孪生兄弟"多带了"往哪个方向"的信息。
Scalar or vector? · 标量还是矢量？
Sort each quantity by whether it needs a direction. · 根据是否需要方向对每个物理量进行排序。
Which of these is a vector? · 以下哪项是矢量？
Velocity has a size and a direction, so it is a vector. Temperature, mass, and time are scalars. · 速度既有大小又有方向，因此是矢量。温度、质量和时间是标量。
Select all · 所有 scalars. · 选出所有标量。
Speed and energy are scalars (size only). Force and displacement are vectors. · 速率和能量是标量（仅有大小）。力和位移是矢量。
Adding vectors
- Vectors add tip-to-tail, not just by numbers.
- The easiest method: break each into components 分量 along the axes, then add the components.
- Two forces of $3\ \text{N}$ can sum to anything from $0$ to $6\ \text{N}$, depending on their directions.
矢量相加
- 矢量首尾相接地相加,而不只是按数值相加。
- 最简单的方法:把每个矢量沿坐标轴分解成分量,再把分量相加。
- 两个 $3\ \text{N}$ 的力,根据它们的方向,合力可以是 $0$ 到 $6\ \text{N}$ 之间的任何值。
You walk $9\ \text{km}$ east then $12\ \text{km}$ north. How far are you from the start (in km)? · 你向东走$9\ \text{km}$，然后向北走$12\ \text{km}$。你距离起点多远（单位：km）？
The displacement magnitude is $\sqrt{9^2 + 12^2} = \sqrt{225} = 15\ \text{km}$ -- not $21$. · 位移的大小是$\sqrt{9^2 + 12^2} = \sqrt{225} = 15\ \text{km}$——而不是$21$。
Two $4\ \text{N}$ forces always add up to $8\ \text{N}$. · 两个$4\ \text{N}$力总是相加得到$8\ \text{N}$。
Only if they point the same way. At right angles they give $\sqrt{4^2+4^2} \approx 5.7\ \text{N}$; opposite, $0$. · 仅当它们指向同一方向时成立。成直角时给出$\sqrt{4^2+4^2} \approx 5.7\ \text{N}$；相反方向时，$0$。
Magnitude and direction
- From components $(A_x, A_y)$, the magnitude 大小 is $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$.
- The direction is the angle $\theta = \tan^{-1}(A_y/A_x)$.
- Components and magnitude-angle are just two languages for the same vector.
大小与方向
- 由分量 $(A_x, A_y)$,大小为 $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$。
- 方向是角度 $\theta = \tan^{-1}(A_y/A_x)$。
- 分量表示和"大小-角度"表示,只是同一个矢量的两种语言。
A vector has components $A_x = 6$, $A_y = 8$. What is its magnitude? · 一个矢量的分量为$A_x = 6$，$A_y = 8$。其模长是多少？
$|\vec{A}| = \sqrt{6^2 + 8^2} = \sqrt{100} = 10$.
A vector has components $A_x = 3$, $A_y = 4$.
- Magnitude: $|\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
- Direction: $\theta = \tan^{-1}(4/3) \approx 53^\circ$ above the x-axis.
一个矢量的分量为 $A_x = 3$,$A_y = 4$。
- 大小:$|\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$。
- 方向:$\theta = \tan^{-1}(4/3) \approx 53^\circ$,在 x 轴上方。
You cannot add vectors like plain numbers. Walking $3\ \text{km}$ east then $4\ \text{km}$ north leaves you $5\ \text{km}$ away -- not $7$. Only when two vectors point the same way do their magnitudes simply add.
你不能像普通数字那样相加矢量。向东走 $3\ \text{km}$ 再向北走 $4\ \text{km}$,你离起点 $5\ \text{km}$——不是 $7$。只有当两个矢量指向同一方向时,它们的大小才简单相加。
A scalar has size only; a vector has size and direction. Add vectors by components ($A_x$, $A_y$), then recover the magnitude $\sqrt{A_x^2 + A_y^2}$ and angle. Distance and speed are scalars; displacement, velocity, and force are vectors.
标量只有大小;矢量既有大小又有方向。通过分量($A_x$、$A_y$)相加矢量,再求出大小 $\sqrt{A_x^2 + A_y^2}$ 和角度。距离和速率是标量;位移、速度和力是矢量。