Simpler explanation of Vectors and Scalars

[nae]

Hi everyone! I am currently getting myself started in Physics. But for some reason I can't understand what vectors and scalers actually are.

[swansont]

20 minutes ago, nae said:

Hi everyone! I am currently getting myself started in Physics. But for some reason I can't understand what vectors and scalers actually are.

A scalar is something that can be represented by a number (and usually units) Mass, for example. A rock with a mass of 2 kg.

A vector has a direction. It has a value, but also tells which way. Velocity, for example. “moving 10 m/s in the x-direction”

The magnitude of a vector is a scalar 

“I am 100m north of you” represents a vector. “I am 100m away” represents a scalar.

[studiot]

19 minutes ago, nae said:

Hi everyone! I am currently getting myself started in Physics. But for some reason I can't understand what vectors and scalers actually are.

 

Good for you +1 for encouragement and asking about things.

You will find in Physics that you don't meet everything at once, but learn a little about something and then come back again later to learn some more.

So I will not tell you all about scalars and vectors, just some stuff to hold onto that may help in diferent situations.

Scalars and vectors are part of a heirarchy of types of things (objects) in Physics that are related to coordinate systems.

Are you familiar with the idea of coordinates ?

 

As they are objects they have some special properties.

At the bottom of the list we find scalars.

A scalar has a magnitude or simple numeric value. (Note this could be zero. zero is a perfectly respectable and valid numeric value)
This is the same wherever you are, I see swansont has offered mass as an example.

So a mass of 1 kilogramme is the same in Timbuctoo as Toronto.

Next on the list we have vectors.

These not only have a magnitude but they also have a direction.
The direction is the link that ties into the coordinate system

So we can represent the distance from Toronto to Timbuctoo as a vector.
This will tell us not only how far we have to go (magnitude) but also in what direction we need to set off in order to reach the destination.

 

That is enough to be going on with until you confirm about coordinate.

 

I will mention that the next object in the heirarchy is called a Tensor, in case you hear of them.
A tensor has magnitude and not one but two directions.
But don't worry about them.
 

 

[mathematic]

From a math point of view, a scalar is something that is one dimensional, while a vector is something with more than one (usually two or three) dimensions.

There are rules for combining them.  For example, scalars use ordinary arithmetic.   Addition or subtraction of vectors is straightforward but multiplication is more complicated.

Edited September 2, 2020 by mathematic

[nae]

22 hours ago, studiot said:

 

Good for you +1 for encouragement and asking about things.

You will find in Physics that you don't meet everything at once, but learn a little about something and then come back again later to learn some more.

So I will not tell you all about scalars and vectors, just some stuff to hold onto that may help in diferent situations.

Scalars and vectors are part of a heirarchy of types of things (objects) in Physics that are related to coordinate systems.

Are you familiar with the idea of coordinates ?

 

As they are objects they have some special properties.

At the bottom of the list we find scalars.

A scalar has a magnitude or simple numeric value. (Note this could be zero. zero is a perfectly respectable and valid numeric value)
This is the same wherever you are, I see swansont has offered mass as an example.

So a mass of 1 kilogramme is the same in Timbuctoo as Toronto.

Next on the list we have vectors.

These not only have a magnitude but they also have a direction.
The direction is the link that ties into the coordinate system

So we can represent the distance from Toronto to Timbuctoo as a vector.
This will tell us not only how far we have to go (magnitude) but also in what direction we need to set off in order to reach the destination.

 

That is enough to be going on with until you confirm about coordinate.

 

I will mention that the next object in the heirarchy is called a Tensor, in case you hear of them.
A tensor has magnitude and not one but two directions.
But don't worry about them.
 

 

I am familiar with coordinate's but I haven't fully familiarised myself with it though. Your's and mathematic's helped me further understand swansont's explanation. Thank you!

23 hours ago, swansont said:

A scalar is something that can be represented by a number (and usually units) Mass, for example. A rock with a mass of 2 kg.

A vector has a direction. It has a value, but also tells which way. Velocity, for example. “moving 10 m/s in the x-direction”

The magnitude of a vector is a scalar 

“I am 100m north of you” represents a vector. “I am 100m away” represents a scalar.

Ur explanation is way better than in my textbook and much more simpler compared to it. Thank you!

[studiot]

3 hours ago, nae said:

I am familiar with coordinate's but I haven't fully familiarised myself with it though. Your's and mathematic's helped me further understand swansont's explanation. Thank you!

Thank you for your confidence.

I must go tonight, but I will draw some (hopefully helpful) diagrams tomorrow.

[studiot]

On 9/2/2020 at 7:42 PM, swansont said:

A scalar is something that can be represented by a number (and usually units) Mass, for example. A rock with a mass of 2 kg.

A vector has a direction. It has a value, but also tells which way. Velocity, for example. “moving 10 m/s in the x-direction”

The magnitude of a vector is a scalar 

“I am 100m north of you” represents a vector. “I am 100m away” represents a scalar.

Quote

Joe Kolecki,  NASA

Beginnings

At the heart of all mathematics are numbers.

If I were to ask how many marbles you had in a bag, you might answer, “Three.” I would find your answer perfectly satisfactory. The ‘bare’ number 3, a magnitude, is sufficient to provide the information I seek.

If I were to ask, “How far is it to your house?” and you answered, “Three,” however, I would look at you quizzically and ask, “Three what?” Evidently, for this question, more information is required. The bare number 3 is no longer sufficient; I require a ‘denominate’ number – a number with a name.

Suppose you rejoindered, “Three km.” The number 3 is now named as representing a certain number of km. Such numbers are sometimes called scalars. Temperature is represented by a scalar. The total energy of a thermodynamic system is also represented by a scalar.

If I were next to ask “Then how do I get to your house from here?” and you said, “Just walk three km,” again I would look at you quizzically. This time, not even a denominate number is sufficient; it is necessary to specify a distance or magnitude, yes, but in which direction ?

“Just walk three km due north.” The denominate number 3 km now has the required additional directional information attached to it. Such numbers are called vectors.

Velocity is a vector sinceit has a magnitude and a direction; so is momentum. Quite often, a vector is represented by components. If you were to tell me that to go from here to your house I must walk three blocks east, two blocks north, and go up three floors, the vector extending from “here” to “your house”would have three spatial components:

Three blocks east,

Two blocks north,

Three floors up.

Physically, vectors are used to represent locations, velocities, accelerations, flux densities, field quantities, etc. The defining equations of the gravitational field in classical dynamics (Newton’s Law of Universal Gravitation), and of the electromagnetic field in classical electrodynamics (Maxwell’s four equations) are all given in vector form. Since vectors are higher order quantities than scalars, the physical realities they correspond to are typically more complex than those

represented by scalars.

Swansont and Joe have both said more or less the same thing and given some examples.

In particular there is a heirarchy or ladder; I have pointed out that vectors are not the top of the ladder.

......

Vectors

Scalars

Numbers

So what are the differences and how do you tell these differences?

It is a little more tricky than at first sight, as Joe has hinted since distance can be a scalar or a vector, depending upon circumstances.

But let us start at the bottom.

Numbers and scalars.

Not all numbers are scalars.

3 and 5 by themselves are just numbers.

You can add them together  3  +  5  =  8

Give them units eg 3 apples, 5 oranges.

You cannot add these together

3 apples plus 5 oranges is always 3 apples plus 5 oranges (unless we rename them both fruit).

You can, however add 3 apples and 5 apples to get 8 apples.

So sometimes you can add scalars  - but only if they ahae the same units and they apply to the same situation.

4 kg plus 4 kg is only 8 kg if they are on the same side of the scales.

You can always add two any numbers to obtain a sum number without restriction

Note swansont said scalars usually have units.

Sometimes a scalar can be just a number.
This is, in fact, where the name comes from,
An enlargement of 1.2 or 120% on you photocopier is just a number, as is the scale factor on a map

But adding the scale of one map to the scale of another makes no sense.
And if you enlarge by 1.2 twice you do not get an enlargement of (1.2 + 1.2 =2.4) you get an enlargement of (1.2 * 1.2 = 1.44)

So a number can be a scalar (one step up the ladder) or it can be just a number.

Similarly Joe has noted that distance can be a scalar, with units, or it can be one step up the ladder as a vector.

 

Talking of maps neatly brings us to coordinate systems where we can start to look at the differences between scalars and vectors.

 

On 9/3/2020 at 6:55 PM, nae said:

I am familiar with coordinate's but I haven't fully familiarised myself with it though.

 

I assume you are familiar with plotting a graph of a function  ?

That is a form of coordinate system. We do not need fancy coordinates systems, a simple X - Y grid as in Fig 1will suffice.

The important points are that X and Y are variables that on take on any value  - that is we can choose any value for Y without regard to the value of X.
So we can place A, B and C where we like on the grid.

We say that X and Y are both independent variables and we need (at least) two of them to create a vector.
Before we come to that,

The graph plotted in fig 2 relates to swansont's example of speed and velocity.

Both speed and power are scalars.

The speed of our vehicle depends upon the power supplied by the engine.

So we cannot place A, B and C anywhere on the XY plot we are constrained by the relationship between X and Y

In this case we say that We say that X is an independent variable  and Y is a dependent variables

 

We will examine the significance of this when we come to plot vectors in the next instalment, along with the significance of the units of scalers for the addition of vectors.

 

How are we doing?

[nae]

19 hours ago, studiot said:

How are we doing?

That was an really good explanation. I don't know who u are but u should become a teacher or something(if u aren't already). I feel like I understand Scalers and Vectors a lot more than I did before mostly cause fig 1 and 2.  If u don't mind I'm going to use some of ur explanations in my notes since it was nicely simplified. Once again thank you! ❤️