Scalars and vectors · 标量和矢量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| distance/ˈdɪstəns/ | 距离 | jù lí |
| displacement/dɪˈspleɪsmənt/ | 位移 | wèiyí |
| scalar/ˈskeɪlə/ | 标量 | biāoliàng |
| vector/ˈvektə/ | 矢量 | shǐliàng |
| density/ˈdensɪti/ | 密度 | mìdù |
| resultant/rɪˈzʌltənt/ | 合矢量 | héshǐliàng |
| component/kəmˈpəʊnənt/ | 分量 | fēnliàng |
| perpendicular/ˌpɜːpənˈdɪkjʊlə/ | 垂直 | chuízhí |
Which way?
- Walk $3\ \text{km}$ out and $3\ \text{km}$ back: you travel $6\ \text{km}$, but end up where you started.
- The distance 距离 is $6\ \text{km}$; the displacement 位移 is $0$.
- The difference is direction — and that is the whole idea here.
朝哪个方向?
- 走出去 $3\ \text{km}$ 再走回来 $3\ \text{km}$:你走了 $6\ \text{km}$,但回到了出发点。
- 距离(distance) 是 $6\ \text{km}$;位移(displacement) 是 $0$。
- 区别在于方向——这正是这里的核心。
Scalars 标量 and vectors 矢量
- A scalar has size only.
- A vector has size and direction.
The SI prefixes climb in steps of a thousand, from pico to giga
标量和矢量
- 标量(scalar) 只有大小。
- 矢量(vector) 既有大小又有方向。
Resolving forces on a slope · 在斜面上分解力
A weight on a slope resolves into a component down the slope (mg·sinθ) and one into it (mg·cosθ); the block slides when mg·sinθ beats friction. · 斜面上的重量分解为沿斜面向下的分量(mg·sinθ)和垂直压入斜面的分量(mg·cosθ);当 mg·sinθ 超过摩擦力时,物块滑动。
Scalars & vectors · 标量与矢量
resultant = a + b · 合矢量 = a + b
A vector has direction; add two tip-to-tail to get the resultant. · 矢量 有方向;把两个 首尾相接 相加得到合矢量。
You walk $4\ \text{km}$ east, then $4\ \text{km}$ back west. What is your displacement, in km? · 你向东走 $4\ \text{km}$,再向西走回 $4\ \text{km}$。你的位移是多少(单位 km)?
Displacement is a vector — start to finish. You end where you began, so it is $0$. The distance travelled is $8\ \text{km}$. · 位移是 矢量——从起点到终点。你回到了出发点,所以是 $0$。走过的距离是 $8\ \text{km}$。
Which is which?
- Scalars: mass, time, temperature, energy, work, power, distance, speed, density 密度.
- Vectors: displacement, velocity, acceleration, force (incl. weight), momentum.
哪个是哪个?
- 标量: 质量、时间、温度、能量、功、功率、距离、速率、密度。
- 矢量: 位移、速度、加速度、力(包括重力)、动量。
Select all · 所有 the quantities that are vectors. · 选出所有是 矢量 的量。
Vectors have a direction: displacement, velocity, force and momentum. Mass, energy and speed are scalars (speed is the size of velocity, with no direction). · 矢量有方向:位移、速度、力和动量。质量、能量和速率是标量(速率是速度的大小,没有方向)。
Which of these are vectors? Select all · 所有 that apply. · 下列哪些是矢量?选出所有适用的。
Energy is a scalar, which is why kinetic energies simply add in a collision while momenta must be added as vectors with signs. · 能量是标量,这就是碰撞中动能可以直接相加、而动量必须带符号作矢量相加的原因。
A quick test
- Ask: "in which direction?"
- "In which direction is the temperature?" makes no sense → temperature is a scalar.
- "In which direction is the velocity?" makes sense → velocity is a vector.
一个快速检验
- 问:“朝哪个方向?”
- “温度朝哪个方向?”说不通 → 温度是 标量。
- “速度朝哪个方向?”说得通 → 速度是 矢量。
Temperature is a vector. · 温度是矢量。
You cannot ask "in which direction is the temperature?", so it is a scalar. · 你不能问“温度朝哪个方向?”,所以它是 标量。
Vectors are arrows
- Draw a vector as an arrow.
- Its length (to scale) shows the magnitude; its direction shows the direction.
A vector is an arrow — its scaled length is the magnitude and the way it points is the direction
矢量是箭头
- 把矢量画成一个箭头。
- 它的 长度(按比例)表示大小;它的 方向 表示方向。
一个矢量就是一个箭头——它按比例的长度是大小,它指向的方式是方向
Adding vectors: tip to tail
- Draw the arrows tip to tail.
- The resultant 合矢量 goes from the first tail to the last tip.
矢量相加:首尾相接
- 把箭头 首尾相接(tip to tail) 地画。
- 合矢量(resultant) 从第一个箭头的尾画到最后一个箭头的头。
Two forces act at a point: $3\ \text{N}$ east and $4\ \text{N}$ north. What is the size of the resultant? · 两个力作用在一点:$3\ \text{N}$ 向东,$4\ \text{N}$ 向北。合力的大小是多少?
They are at right angles, so $|R| = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5\ \text{N}$. · 它们成直角,所以 $|R| = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5\ \text{N}$。
Subtracting vectors
- To find $\vec{X} - \vec{Y}$, add the reverse of $\vec{Y}$: $\vec{X} + (-\vec{Y})$.
- The reverse has the same size but points the opposite way.
矢量相减
- 求 $\vec{X} - \vec{Y}$,就加上 $\vec{Y}$ 的 反向:$\vec{X} + (-\vec{Y})$。
- 反向矢量大小相同,但指向相反。
To find $\vec{X} - \vec{Y}$ by drawing arrows, you: · 用画箭头的方法求 $\vec{X} - \vec{Y}$,你要:
$\vec{X} - \vec{Y} = \vec{X} + (-\vec{Y})$ — reverse $\vec{Y}$ (same size, opposite direction) and add tip to tail. · $\vec{X} - \vec{Y} = \vec{X} + (-\vec{Y})$——把 $\vec{Y}$ 反向(大小相同、方向相反)再首尾相接相加。
Resolving into components 分量
- Any vector splits into two perpendicular 垂直 parts.
- For a vector $v$ at angle $\theta$ to the horizontal: $v_{\text{H}} = v\cos\theta$, $v_{\text{V}} = v\sin\theta$.
分解成分量
- 任何矢量都可以分解成两个 互相垂直 的部分。
- 对于与水平方向成角 $\theta$ 的矢量 $v$:$v_{\text{H}} = v\cos\theta$,$v_{\text{V}} = v\sin\theta$。
A velocity of $10\ \dfrac{\text{m}}{\text{s}}$ points $60^{\circ}$ above the horizontal. What is its horizontal part? · 一个速度 $10\ \dfrac{\text{m}}{\text{s}}$ 指向水平线上方 $60^{\circ}$。它的水平分量是多少?
Horizontal part $= v\cos\theta = 10 \times \cos 60^{\circ} = 10 \times 0.5 = 5\ \dfrac{\text{m}}{\text{s}}$. · 水平分量 $= v\cos\theta = 10 \times \cos 60^{\circ} = 10 \times 0.5 = 5\ \dfrac{\text{m}}{\text{s}}$。
Put the method for resolving a vector on a slope in order. · 把在斜面上分解矢量的方法按顺序排列。
Choosing axes along the slope is what makes one component zero. Guessing between sine and cosine is the single commonest error with vectors. · 取沿斜面的轴,正是让某个分量为零的办法。在正弦和余弦之间靠猜,是矢量题最常见的错误。
A 50 N force acts at 30 degrees to the horizontal. What is its horizontal component, in N? · 一个 50 N 的力与水平成 30 度角。它的水平分量是多少 N?
The angle is measured from the horizontal, so the horizontal component is the adjacent side: 50 cos 30 = 43.3 N. An angle given from the VERTICAL flips sine and cosine. · 角是从水平方向量起的,所以水平分量是邻边:50 cos 30 = 43.3 N。若角是从竖直方向量起,正弦和余弦就要对调。
On a slope
- On a slope of angle $\theta$, split the weight along and across the slope.
- Along: $W_{\parallel} = W\sin\theta$. Across: $W_{\perp} = W\cos\theta$.
在斜面上
- 在角度为 $\theta$ 的斜面上,把重力沿斜面和垂直斜面分解。
- 沿斜面:$W_{\parallel} = W\sin\theta$。垂直斜面:$W_{\perp} = W\cos\theta$。
On a slope of angle $\theta$, the part of the weight acting along the slope is $W$ ____ $\theta$. · 在角度为 $\theta$ 的斜面上,重力 沿斜面 作用的分量是 $W$ ____ $\theta$。
Along the slope, $W_{\parallel} = W\sin\theta$; the part pressing into the slope is $W_{\perp} = W\cos\theta$. · 沿斜面方向,$W_{\parallel} = W\sin\theta$;压入斜面的分量是 $W_{\perp} = W\cos\theta$。
When do you resolve?
- Whenever you need how much of a vector acts in one direction.
- Example: the part of a force pulling along a ramp, or the up and across parts of a thrown ball's velocity.
什么时候要分解?
- 只要你需要某个矢量在某一方向上有多少。
- 例如:一个力沿斜坡拉动的部分,或一个被抛出的球的速度在向上和水平方向的部分。
You've got it
- a vector has direction, a scalar does not (ask "in which direction?")
- add vectors tip to tail; at right angles $|R| = \sqrt{X^{2}+Y^{2}}$
- resolve a vector with $v\cos\theta$ (horizontal) and $v\sin\theta$ (vertical)
你掌握了
- 矢量 有方向,标量 没有(问“朝哪个方向?”)
- 矢量 首尾相接 相加;成直角时 $|R| = \sqrt{X^{2}+Y^{2}}$
- 用 $v\cos\theta$(水平)和 $v\sin\theta$(竖直)分解 一个矢量