Scalars vs Vectors

Why is current a scalar? (preview)

0:00
Point it 37.5 degrees south of east. If that sounds strange, that direction is fake, so it's a scalar. If it sounds natural, the direction is real. Say I'm moving 37.5 degrees south of east. You'd think: fine. Velocity's direction is real. Now current: 37.5 degrees south of east? Awkward. Current's direction is fake

Why vectors and scalars matter

0:25
In this video: vectors and scalars. This is crucial for high school physics, but teachers often skip it, and students tend to brush it off. By the end, you'll master four points. First: how to represent vectors and scalars. Second: writing a 1D vector simply. In high school problems, 1D vectors are what's tested most. No complicated 2D or 3D vectors. Third: real and fake directions. Fourth: reading plus/minus signs right. Starting with part one.

Vectors use arrows, scalars use numbers

1:01
How to represent vectors and scalars. Vectors first: a quantity with magnitude and direction. With direction, a plain number won't do. How do we show direction? Simple: draw an arrow. Common vectors: velocity and force. Here's an example. A box lies on a horizontal surface, and a person, say Qi Xiaoyu, pushes it from left to right. To push it, she must exert a force on it. How do we show that force? First, it points right. So I draw an arrow pointing right. She can push hard or push gently. How do we show which? Easy. A big push gets a long arrow, a small push gets a short arrow. A vector is drawn as an arrow; its length shows the magnitude, e.g. of a force. The arrow's direction is the vector's.

2:06
A right arrow means a left-to-right push. Velocity and force again. We skip velocity today, Lesson 6 covers it: its direction, its magnitude, and so on. That's vectors. Now scalars, which are simpler. A scalar has only magnitude, no direction, so one number is enough. Common ones: mass, volume, temperature. Speaking of mass, Nobody says 'mass to the east' or west. Same for volume: magnitude, no direction. That's a scalar: written as one number.

Simple representation of a 1D vector

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We said a vector is drawn as an arrow. Isn't drawing arrows a hassle? When things are a hassle, make a habit of asking first: can I be lazy here? Yes, there is. In high school physics, most of what we meet is 1D vectors. Take kinematics: it's a cart on a straight road, going left or right. Two directions. Or on a slope: going up or sliding down. Again, just two directions. A vector with only two directions is called a 1D vector. It has a very simple notation, used all the time in high school physics. For example: Say I give you a coordinate axis, a straight line. An object moves in a line on the axis, left or right, in a line. Let's find a simpler way to write an object's displacement. How do we do that? Next lesson explains displacement. For now: displacement is a vector.

3:48
Its direction points from start to end. Say I walk from O to A. The displacement points right. Its magnitude is the length from start to end. From O to A, the magnitude is 3 m. To draw the displacement from O to A, I draw an arrow pointing right, with a length of three units. That is the full displacement. Now, this problem has a restriction. The object can only move along this line, either left or right. Every displacement arrow we draw points left or right. O to A is this arrow. O to B is this arrow. I can draw a whole pile, but all of them point left or right. Now I can be lazy. How? I fix a direction called the positive direction. I just pick one of the two. Say I pick right, the positive direction. Left, the opposite, is negative.

4:52
Now every arrow pointing right gets a plus sign, and its magnitude is still its length. Take O to A. It points right, the positive direction, so the displacement is positive. Its magnitude is 3 units, which is 3 m. So the displacement from O to A can be written x(OA) = +3 m. See? The arrow is now just one number. Lazy, right? That's the sign convention. But note: this works only for 1D vectors. A 2D vector, like planar motion, isn't limited to a line. Endless directions. Plus and minus signs aren't enough. It worked before as we had two directions. You can count them. In 2D, you can't. That's the simple way to write a 1D vector, and it's extremely common. From now on, whenever a problem writes a vector as one number, like a force of +5 N or -2 N, or a velocity of +5 m/s or -3 m/s, it's doing exactly this: the sign convention.

6:08
A very common notation in physics. Once we've covered 1D vector notation,

Why current is a scalar: a memory aid (real vs. fake direction)

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vectors and scalars are done. But that alone isn't enough for much of what comes later. For example, when I learned electric current at school, my teacher said current's direction is that of positive charge: + to −. Easy. A direction, so it's a vector, I thought. But then: "Note: current is a scalar. No direction." I was instantly stunned. First you say current has a direction: positive to negative. then that it has none? It's a scalar. So which is it? I'm sure you'll wonder the same thing. I spent days puzzling over it. I finally got it. What follows is my own understanding, maybe not rigorous, but a big help. Direction needs a finer split. Some directions are real, some are fake. A real direction? A direction in space, east, west, south or north, like the direction you run.

7:22
Those are real. And fake directions? A vivid example from studying: you work hard toward a class rank, toward passing, toward a score of 90. Is that east, west, south or north? You can't say. It's still called a direction, but it has no compass bearing, so it's a fake direction. Current's direction, positive to negative, is also fake. Say current points east, west, south, north? Sounds odd. So it's fake. So don't assume all scalars have no direction. Think so, and much later won't add up. So how should we understand scalars? Important: some scalars do have a direction, but it's always a fake direction. Only vectors have real directions. So to tell them apart: a vector always has a direction, always a real one. Scalars: some have none, like mass and volume.

8:26
Others have a direction, but a fake one. From now on, hold three things in mind: real direction, fake direction, and no direction. No more doubts. When current's direction comes up, it'll make sense. Here's a handy trick. Say you learn a new quantity, then later forget whether it has a direction, whether it's a vector. Try this: Point it 37.5 degrees south of east. If that sounds strange, that direction is fake, so it's a scalar. If it sounds natural, the direction is real. Say I'm moving 37.5 degrees south of east. You'd think: fine. Velocity's direction is real. Now current: 37.5 degrees south of east? Awkward. Current's direction is fake. Or: my effort points 37.5 degrees south of east? Makes no sense. Weird. Now you know real and fake directions. With these two ideas, directional scalars now make sense.

Three uses of plus and minus signs

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Last: how to read plus and minus signs. Signs have three uses. Hear them once, avoid many problems later. First use: the sign shows which is bigger. Take temperature: positive beats negative. Use one: which is bigger. Same as the signs in math: it shows size. Second: a 1D vector's direction, That's the sign convention. Plus: same as positive direction, minus: opposite. Note: the sign isn't used to compare magnitude. Example: an object on a flat surface. It lies there. Take right as positive. Qi Xiaoyu first pushes it to the right, with a force F1 = 3 N. The second push is to the left, with F2 = 10 N. With the sign convention, F1 = +3 N, and F2 = -10 N, as F1 is along the positive direction, F2 opposite. So which is bigger, F1 or F2?

10:35
Obviously F2 is bigger. So when comparing vectors, we ignore signs and look only at numbers. When a sign shows a 1D vector's direction, remember: compare numbers, not signs. Third: it shows a property's two values. If a property has only two values, use plus and minus. Say a simple game has only two classes, a mage and a warrior. Plus can mean warrior, minus mage. In real life, boys and girls: plus boys, minus girls, or the reverse. Either works. The classic case: a power source's terminals. A power source has only two polarities. So we simply use plus and minus. Positive: current flows out; negative: current returns. Here the sign shows no size or direction. It simply names the two polarities: one plus, one minus. Those are the three uses of signs.

11:38
How to understand them? Just one sentence: don't overthink it. Treat them as an ordinary pair of symbols that carry some information. A sign needn't show size, like a power source's terminals. And not only vectors have signs. Early on, the quantities students learn are vectors with signs, like displacement +5 m, -3 m, velocity +3 m/s, -2 m/s, force -3 N, +2 N: all these vectors carry signs. A mental habit forms: only vectors can have signs. Every year, some students, when they later learn a scalar that has a sign, ask: it's a scalar, so why a sign? It's that habit. So don't read too much into signs. Just a pair of symbols with three uses. Did you nail the four points?

Summary: the four points

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First, vectors and scalars: Vector: arrow. Scalar: number. Clear? Second, a simple form for 1D vectors: a number with a sign. Note: the sign isn't used to compare magnitude. Third and fourth: my secret stash, saved just for you. Real and fake directions: Running direction is real: N, S, E, W. Your direction of progress, of effort, is a fake direction. It has no compass points. A classic fake direction is current. This explains why some scalars have a direction. They're all fake. A scalar can't have a real direction. A real direction is the key to telling vector from scalar. Last, signs. The core: don't overthink. Treat them as an ordinary pair of symbols for information. Simple and easy to write. They needn't show size, nor belong only to vectors. Three uses!