Scalars and Vectors in One Dimension
Introduced| English | Meaning |
|---|---|
| direction/daɪˈrekʃn/ | direction |
| scalar/ˈskeɪlə/ | A quantity described by magnitude alone, with no direction. |
| magnitude/ˈmæɡnɪtjuːd/ | The size or numerical value of a quantity. |
| speed/spiːd/ | The distance travelled per unit time. |
| vector/ˈvektə/ | A quantity with both magnitude and direction. |
| displacement/dɪˈspleɪsmənt/ | The straight-line change in position, including direction. |
| velocity/vəˈlɒsɪti/ | The rate of change of displacement. |
| positive direction/ˈpɒzɪtɪv daɪˈrekʃn/ | The direction chosen to have positive coordinate or vector-component values. |
| resultant/rɪˈzʌltənt/ | The single vector equal to the sum of two or more vectors. |
"It moved 3 meters." Which way?
- A rescue drone is told: "the hiker is 3 kilometers away."
- Away in which direction? North? Up a cliff? The number alone can't guide the drone.
- Some quantities need only a size. Others are useless without a direction too.
- Kinematics starts by sorting every quantity into one of these two kinds.
Scalars: size only
- A scalar 标量 has only a magnitude 大小 — a number with a unit.
- No direction is attached, and none is needed.
- Examples: time (5 s), mass (2 kg), temperature (20 °C), speed 速率 (30 m/s), energy, distance.
- You can add scalars with ordinary arithmetic: 2 kg + 3 kg = 5 kg.
Vectors: size and direction
- A vector 矢量 has a magnitude and a direction 方向.
- Change the direction and you have a different vector, even if the size is the same.
- Examples: displacement 位移, velocity 速度, acceleration, force.
- "$30\ \tfrac{\text{m}}{\text{s}}$ east" is a vector; "$30\ \tfrac{\text{m}}{\text{s}}$" on its own is just a speed.
Scalar or vector?
Sort each quantity into scalar (size only) or vector (size and direction).
Which of these is a vector quantity?
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Velocity has a size and a direction, so it is a vector. Mass, temperature and speed are scalars (size only).
A vector has a magnitude and a ____.
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A vector is defined by both a magnitude and a direction.
On a line, a sign is a direction
- In one dimension we don't need arrows or angles — just a positive direction 正方向.
- A positive value points one way; a negative value points the opposite way.
- So $+3\ \text{m}$ and $-3\ \text{m}$ have the same size but opposite directions.
Select all the quantities that are vectors.
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Displacement, force and velocity all need a direction. Distance and temperature are scalars.
A velocity of $-8\ \tfrac{\text{m}}{\text{s}}$ is not "slower" than $+8\ \tfrac{\text{m}}{\text{s}}$. They have the same speed — the minus sign only means it points the other way.
A velocity of $-8\ \tfrac{\text{m}}{\text{s}}$ is slower than a velocity of $+8\ \tfrac{\text{m}}{\text{s}}$.
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Both have the same speed ($8\ \tfrac{\text{m}}{\text{s}}$). The minus sign is a direction, not a smaller size.
Adding vectors on a line
- To combine vectors along a line, just add the signed numbers.
- A walk of $+5\ \text{m}$ then $-2\ \text{m}$ gives a resultant 合矢量 of $+3\ \text{m}$.
- The total distance walked is $7\ \text{m}$, but the displacement is only $+3\ \text{m}$.
- This is why displacement (a vector) and distance (a scalar) can be very different.
A walk of $+5\ \text{m}$ followed by $-2\ \text{m}$ gives a displacement of how many metres?
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Add the signed values: $(+5) + (-2) = +3\ \text{m}$. The total distance is $7\ \text{m}$, but the displacement is only $+3\ \text{m}$.
Put these distances in order, smallest first.
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Ordering by magnitude builds your sense of scale: $0.7\ \text{m}$, then $12\ \text{m}$, then $400\ \text{m}$.
A cyclist rides 300 m east, then 100 m west.
- Distance travelled (scalar) $= 300 + 100 = 400\ \text{m}$.
- Displacement (vector) $= (+300) + (-100) = +200\ \text{m}$ — that is, 200 m east.
Same trip, two very different answers: one counts every metre, the other counts only the change in position.
Scalar = magnitude only (time, mass, speed). Vector = magnitude and direction (displacement, velocity). In one dimension, a + / − sign carries the direction.
Write from a displacement of 5 m
- Walk +9 m and then -4 m along a line.
- Find the displacement. The path length is not the answer.
A walk of +9 m is followed by -4 m along the same line. What is the displacement, in metres?
9 minus 4 is 5.
What is the displacement?
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Signed addition of 9 and negative 4 is 5 m. Other options — 13 m (the path length); 3 m (the lesson displacement).