Lesson 4 of 10
Scalars vs Vectors
矢量与标量
Current has a direction, so why is it a scalar?
Lesson notes
A short written version of Lesson 4. The study sheet has more practice questions.
Short answer
Current is a scalar because its direction is a fake direction. "From the positive to the negative pole" is not a direction in space. Only vectors have real directions.
Key ideas
- A vector has both magnitude and direction. Draw it as an arrow: the length shows the magnitude. Examples: velocity, force.
- A scalar has magnitude only. Write it as a number. Examples: mass, volume, temperature.
- A 1D vector has only two directions. Pick one as positive. Then the vector is a signed number, like x(OA) = +3 m or a force of −2 N.
- To compare vectors, ignore the sign and compare the numbers. A force of 10 N is bigger than 3 N.
- A real direction is a direction in space: east, west, south or north. Some scalars have a direction, but it is always fake.
- The + and − signs have three uses: order of size, the direction of a 1D vector, and the two values of a property.
Step by step
- First ask: does the quantity have a direction? Current does. It flows from the positive to the negative pole.
- Next ask: is that direction real or fake? A real direction is a direction in space.
- Use the quick test. Say "current points 37.5° south of east". It sounds odd.
- Compare velocity. "Moving 37.5° south of east" sounds fine, so the direction of velocity is real.
- So the direction of current is fake. Only vectors have real directions, so current is a scalar.
Common mistakes
- Thinking anything with a direction is a vector. Some scalars, like current, have a direction, but it is always a fake direction.
- Thinking the sign shows size. The sign of a 1D vector shows direction only. Compare the numbers: a force of −10 N is bigger than a force of +3 N.
- Thinking only vectors can have + or − signs. A battery's poles also use + and −, and they are not vectors.
- Writing a 2D vector as one signed number. A 2D vector has endless directions, but + and − can show only two.
Practice
Question 1. Decide whether each direction is real (R), fake (F) or there is no direction (N). (1) the direction of velocity (2) the direction of current (3) mass (4) "I'm aiming for a score of 90". Current flows from the positive to the negative pole, so why is it a scalar?
Show the answer
(1) R (2) F (3) N (4) F. Current: its direction is fake (positive to negative pole, not a direction in space). A scalar can have a fake direction. Only vectors have real directions, so current is a scalar.
Question 2. Qi Xiaoyu pushes a box to the right with F₁ = 3 N, then to the left with F₂ = 10 N. Take right as positive. (a) Write F₁ and F₂ as signed numbers. (b) Which force is bigger? Why?
Show the answer
(a) F₁ = +3 N (same direction as +), F₂ = −10 N (opposite to +). (b) F₂ is bigger. When comparing vectors, ignore the sign and compare the numbers: 10 N > 3 N.
Words in English and Chinese
- vector
- 矢量
- scalar
- 标量
- 1D vector
- 一维矢量
- vector algebraization
- 矢量代数化
- positive direction
- 正方向
- real / fake direction
- 真方向 / 假方向
- displacement
- 位移
- current
- 电流
Sum it up before you move on.
Write this lesson in one line:
- What was the question?
- What is the simple version?
- What would you have got wrong?