What are vectors and scalars?

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I read about it and did some googling, but didn’t really understand it. I was hoping someone here could break it down for me. I’d appreciate both examples and definitions on the terms, as well as of course, an explanation.

In: Mathematics

10 Answers

Anonymous 0 Comments

There are a lot of good explanations here on the difference in physical terms so I’ll add the mathematical part too.

A scalar is a single number. You see these all the time so you probably don’t need an example.

A vector is a set of numbers. They are usually written down as an array. For example:
[3,5,7]
or
[3,
5,
7]
the brackets in the second one usually go all the way from top to bottom but I don’t know how to do that in reddit. Those would be called 3×1 or 1×3 vectors, respectively.

Either a scalar or a vector can express many different things. When we’re working with physical objects we’re generally limited to 1×3 (3×1) or 1×4 (4×1) vectors. Each position would correspond to x,y,z and possibly time.

But we can also use vectors to represent other things. For example, if there are a bunch of candidates, a vector could be how many votes each candidate got in a particular state.

You can take the concept a step further and have a whole matrix. For example:
[3,5,7
6,4,2
3.5, 600,-676]

Those numbers can also correspond to all kinds of things. For example each row and column might represent a person in a class and each number might represent how much that person likes the other person (the diagonal row would probably be something like all 1 or all 0 in this case).

There are also a bunch of rules on how you can do math with scalars, vectors and matrices. That’s well outside the scope of an ELI5 though.

Anonymous 0 Comments

As the formal definition of vectors and scalars have already been mentioned, I’m going to give an easy example.

Vectors can be used to classify things. All kinds of things. Let’s say you want to sell a house. Criteria for advertising are location, size, age, price.

In this model, any House can be described with a vector.

Also, this is an example where dimensions higher than 3 pop up. Our house has 4 parameters, so it’s a 4 dimensional vector.

Anonymous 0 Comments

A scalar is something that only has a size or amount. Examples of this are mass and speed. A person may have a mass of 70 kg, or a car can be travelling at 53 mph. These measurements have a size (magnitude), but there is no direction.

A vector is something that has both a size and a direction. An everyday example of this is the weather forecast that will describe the wind as blowing at 10 mph from the South.

Anonymous 0 Comments

A vector is a magnitude plus a direction. If I’m in my car going 60 kph due north, there’s a magnitude (my speed of 60 kph) and a direction (north). That vector can be canceled out – like let’s say my car is on a big, crazy conveyor belt that’s going 60 kph due south – now those two vectors (call them +60 and -60) cancel each other out, and I don’t move.

A scalar is only a magnitude, with no direction. Mass is an easy one, because if you weigh 75 kg or whatever, there’s no up, down, north, south, or anything about it. It’s just mass. There’s no way to reverse mass to cancel that out like you could with a vector – it’s just a data point. Though if you added in a direction, you could create a vector – like if you factor in gravity, now my mass is exerting a force (G) downward (because gravity pulls down relative to me). If you cancel that out with the same force going up, there’d be no net force acting on me, because +G and -G will cancel out. But even though the vectors would cancel out, that scalar number of my mass is still the same, because scalars don’t have a direction and can’t cancel each other. I’d just be floating with the same mass.

Anonymous 0 Comments

A vector is a measurement that had both a value and a direction. These include things like your motion (10 mph to the right is different than 10 mph to the left) or forces.

Anything that has a convention for both positive and negative numbers is really a vector. Positive means one way, negative means the opposite way. So +$10 in my bank means different things than -$10. One is money moving into it (the direction) the other is money moving out.

Scalar measurements are for things where there is no direction, like weight. You don’t weigh -150 lbs… That doesn’t happen or have meaning that is opposite 150 lbs. Sound volume is another.

Where this is important is how they combine. Moving right +10 mph, while in a train moving left at -10 mph means you aren’t really going anywhere (+10 – 10=0). But for weight, you pick something up you always weigh more.

Anonymous 0 Comments

A vector is something that has a magnitude and a direction. A scalar has magnitude but no direction.

I weigh 220 lbs. That’s a scalar.

Get in your car and drive north for 10 km. That’s a vector – it has a magnitude (10 km) and a direction (north)

Anonymous 0 Comments

Vector and scalar are the way(s) things are measured. Measuring something means counting it. If you look at a numberline:

-2 -1 **0** +1 +2 +3 And so on

Notice from zero, the numbers can be counted backwards and forward.

•In a vector, counting can be left and right, forward, backward, up, and down— vectors can have different directions.

Example: one can drive forward and backward

•A scalar can only go one direction. Only up. Only down. Only left. Only right. Only one single direction meaning only positive or only negative. In the wordy definitions of others, “magnitude” means positive or negative. Negative magnitude means left on the numberline, positive magnitude means right on the numberline.

Example: you can only measure the amount you pee in a day with one type of number, a positive volume. It is biologically impossible to pee backwards.

Someone will get technical and try to nitpick these extremely vague explanations, but that’s the general idea.

Anonymous 0 Comments

If you’re still having trouble visualising the difference between the two, a scalar is like a point/dot in a 2D graph, while a vector is like a line between two points.

Anonymous 0 Comments

If I walk a mile, and then walk another mile, I’ve walked 2 miles. This is a scalar distance. I don’t know how far it will be to go back. I might have walked a mile north and a mile west. Or a mile north and a mile south. All I want to know is how far I’ve travelled.

Picture a modern city. The streets are laid out as a grid. I walk a mile north, a mile east and a mile south. To get back home, I just need to go a mile west. These are vectors. I can add them together by all th north parts together all the East parts together. So if I go 2 blocks north, a block east, a block north and 3 blocks east, it will be the same as if I go 3 blocks north and 4 blocks east.

It tends to be important in physics. A VTOL jet thrusts at an angle. They even call this “vectored” thrust. The downward component pushes against gravity. The backward component pushes the plane forwards.

Anonymous 0 Comments

A “vector” quantity is one that needs a direction to be specified. A “scalar” quantity is one without any direction.

Three examples…

**Speed/Velocity**

Speed is a scalar quantity, but velocity is a vector quantity.

10m/s is a *speed*

10m/s upwards is a *velocity*

(People often use the words “speed” and “velocity” as if they mean the same thing, but they don’t.)

**Force**

Force is measured in newtons (N). It is a vector quantity so it also needs a direction specified.

10N *downwards* is a force.

10N (with no direction specified) is just the size (“*magnitude*”) of a force.

**Temperature**

Direction isn’t really an issue here, so the quantities here would only be scalars.