Scalars and vectors
Introduced| English | Meaning |
|---|---|
| distance/ˈdɪstəns/ | The total length of a path travelled. |
| displacement/dɪˈspleɪsmənt/ | The straight-line change in position, including direction. |
| scalar/ˈskeɪlə/ | A quantity described by magnitude alone, with no direction. |
| vector/ˈvektə/ | A quantity with both magnitude and direction. |
| density/ˈdensɪti/ | Mass per unit volume. |
| resultant/rɪˈzʌltənt/ | The single vector equal to the sum of two or more vectors. |
| component/kəmˈpəʊnənt/ | The part of a vector acting in a specified direction. |
| perpendicular/ˌpɜːpənˈdɪkjʊlə/ | Meeting at a right angle of 90 degrees. |
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.
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
Resolving forces on a slope
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.
Scalars & vectors
resultant = a + b
A vector has direction; add two tip-to-tail to get the resultant.
You walk $4\ \text{km}$ east, then $4\ \text{km}$ back west. What is your displacement, in km?
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Displacement is a vector — start to finish. You end where you began, so it is $0$. The distance travelled is $8\ \text{km}$.
Which is which?
- Scalars: mass, time, temperature, energy, work, power, distance, speed, density 密度.
- Vectors: displacement, velocity, acceleration, force (incl. weight), momentum.
Select all the quantities that are vectors.
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Vectors have a direction: displacement, velocity, force and momentum. Mass, energy and speed are scalars (speed is the size of velocity, with no direction).
Which of these are vectors? Select all that apply.
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Energy is a scalar, which is why kinetic energies simply add in a collision while momenta must be added as vectors with signs.
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.
Temperature is a vector.
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You cannot ask "in which direction is the temperature?", so it is a scalar.
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
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.
Two forces act at a point: $3\ \text{N}$ east and $4\ \text{N}$ north. What is the size of the resultant?
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They are at right angles, so $|R| = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5\ \text{N}$.
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.
To find $\vec{X} - \vec{Y}$ by drawing arrows, you:
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$\vec{X} - \vec{Y} = \vec{X} + (-\vec{Y})$ — reverse $\vec{Y}$ (same size, opposite direction) and add tip to tail.
Resolving into components 分量
- Any vector splits into two perpendicular 垂直 parts.
- For a vector $v$ at angle $\theta$ to the horizontal: $v_{\text{H}} = v\cos\theta$, $v_{\text{V}} = v\sin\theta$.
A velocity of $10\ \dfrac{\text{m}}{\text{s}}$ points $60^{\circ}$ above the horizontal. What is its horizontal part?
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Horizontal part $= v\cos\theta = 10 \times \cos 60^{\circ} = 10 \times 0.5 = 5\ \dfrac{\text{m}}{\text{s}}$.
Put the method for resolving a vector on a slope in order.
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Choosing axes along the slope is what makes one component zero. Guessing between sine and cosine is the single commonest error with vectors.
A 50 N force acts at 30 degrees to the horizontal. What is its horizontal component, in N?
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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.
On a slope
- On a slope of angle $\theta$, split the weight along and across the slope.
- Along: $W_{\parallel} = W\sin\theta$. Across: $W_{\perp} = W\cos\theta$.
On a slope of angle $\theta$, the part of the weight acting along the slope is $W$ ____ $\theta$.
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Along the slope, $W_{\parallel} = W\sin\theta$; the part pressing into the slope is $W_{\perp} = W\cos\theta$.
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.
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)