12 - What are Vectors and Scalars? - Free Educational videos for Students in K-12

12 - What are Vectors and Scalars? - By Math and Science

Transcript


00:00	Hello , Welcome back to Physics . Won The title
00:02	of this lesson is what our scholars and vectors .
00:06	This is probably one of the first topics that students
00:09	get to in physics where it's a really a foreign
00:12	concept . Most of us have never heard of a
00:14	what a scalar is before . You take physics and
00:16	most of us probably don't . We may have heard
00:18	of the word vector , but we really don't know
00:19	what it means until you get to physics . And
00:21	so if you don't understand these things ahead of time
00:23	, you get to physics and sometimes it seems really
00:25	confusing . By the end of this lesson , you
00:27	should understand what a scalar is , what a vector
00:29	is , how they're different and why they're both important
00:32	. And then in the next few sections we're going
00:34	to actually drill down into vectors . And scalar is
00:36	even more and start to learn how to work with
00:38	them . But just to kind of like disarm the
00:40	situation scalar and vector is very , very simple .
00:43	You actually know what they are already . You just
00:44	have never put those words to it before . So
00:47	in in essence , we're gonna talk about scholars first
00:50	. Then on this side of the board , we're
00:51	gonna talk about vectors . Now , a scalar is
00:54	a quantity , any measurement that only has a magnitude
00:58	, it doesn't have any direction information , it's just
01:01	a magnitude . So let's start writing some things down
01:04	. A scaler scale are that's how you write that
01:09	. Its magnitude only now , what I wanna be
01:16	by magnitude only , it means there's no direction ,
01:19	no direction info . Now , once I start giving
01:23	you some examples of scalar , you'll understand why that
01:26	makes total sense . So , some examples of scholars
01:29	would be something like temperature , uh , it would
01:34	be pressure , like air pressure , it would be
01:39	time even though we say time flows . But if
01:43	you think about a moment of time , there's no
01:44	direction to it at that moment is time equal something
01:47	. So that's a scalar quantity Volume of a gas
01:51	, like 34 m3 or something like that . We've
01:54	already talked about this one speed . It tells you
01:57	how fast you're going meters perspective , but it doesn't
01:59	give any direction information . So it's it's a scalar
02:02	and just another example , mass , mass of an
02:04	object . So let's just kind of go through these
02:06	were quickly and understand what that means . So ,
02:09	if you have the temperature , There's a temperature in
02:11	space here where my finger is , it might be
02:14	, you know , 27°C, , right ? And there's
02:16	a temperature over here , 34°C. . Those temperatures are
02:19	different , but there's no directionality to this temperature .
02:22	If I measure the temperature here , which way is
02:24	the temperature pointing this temperature pointing that way , temperature
02:27	pointing that way . No , we know it's just
02:29	not pointing any particular direction . We just know that
02:31	this point in space has a temperature . So it's
02:34	just a number . That's what we call magnitude magazines
02:36	. How big the number is , Right . All
02:39	of these share the same kind of thing . The
02:40	pressure , you might think pressure has a direction ,
02:42	but at this point in space where my finger is
02:44	the pressure of the gas is acting in all directions
02:47	pointing into this point . Right ? So at this
02:50	moment , at this point , the pressure , you
02:52	know , 34 newtons per square meter or whatever the
02:55	pressure is , it's not pointing any particular direction .
02:58	It's a value at a point . Okay , time
03:01	we already talked about volume . If I have a
03:03	volume of gas here in this little space , we
03:05	say maybe it's 34 cubic centimeters , but the volume
03:08	, it's not pointing any direction . It's just a
03:10	value speed we already talked about in mass . This
03:12	marker has a massive , you know , um ,
03:15	point oh five kg , but it's not pointing any
03:18	direction . It's just a value that we give a
03:20	magnitude . So scholars are things you've dealt with all
03:23	your life . They're not pointing any particular direction .
03:26	They're just they're just values , right ? So I've
03:32	already kind of used my finger to gesture here .
03:34	But just as a picture that you can kind of
03:36	keep in your mind , if you want to think
03:37	about a three dimensional space like this , right ?
03:42	This might be X . This might be why this
03:44	might be easy . This is this is my uh
03:47	, kind of a corner of a room showing you
03:49	here is the flat surface . Here's the volume of
03:50	space , X , Y , Z coordinate system .
03:52	Right . Then you might have over here some distance
03:57	are away from the origin . Right ? You might
04:00	be measuring , Let me switch colors to make it
04:03	a little easier to see . At this point in
04:05	space there might be temperature T . So you might
04:07	be measuring it and you know , like I said
04:09	with my finger 50°C or whatever . And then if
04:12	I swing this , this are around pointing any other
04:15	place in space that the tip of this arrow ,
04:18	the this arrow is just representing where I'm measuring it
04:21	. Right ? Um , I can measure here measure
04:23	here , measure here at the tip of the arrow
04:25	, no matter where I pointed . It's just some
04:27	number T . So it's a scalar quantity . The
04:29	temperature is not pointing any particular direction . All right
04:33	now , because these things are um not pointing any
04:37	particular direction . We kind of use regular variable names
04:41	to describe these guys . So scholars are just a
04:43	number . We use regular variables to describe them .
04:52	So in other words , we might use um ,
04:56	T for temperature , right ? T is equal to
04:58	something we might use lower case P for pressure .
05:01	We might use V for volume . We might use
05:04	em for mass . We might use lower case T
05:07	for time or something like this . But the reason
05:09	you might say , well this is kind of stupid
05:11	. Why is he mentioning this ? It's because when
05:13	we have vector quantities , vectors include a direction .
05:16	So vectors are going to have a little arrow above
05:18	the variables to tell you that their vectors . So
05:21	before we kind of get to the vector part ,
05:22	I'm showing you scalar . So just know that when
05:24	you see a variable in a textbook and there's no
05:26	arrow on top . It's just a scalar quantity .
05:28	It's just a number . It's some number , whatever
05:31	value it is at that location , that's what it
05:32	is . Now , let's walk ourselves to the other
05:35	board and talk about the concept of what a vector
05:39	is . I've mentioned it several times in in passing
05:46	in the class because they're so important . But now
05:48	we're finally getting to the actual definition . It's a
05:50	quantity that means quantity that has magnitude , just like
06:00	the scalar does magnitude , but it also has a
06:02	direction . This this this . So when we give
06:07	some examples , you'll understand immediately why it makes sense
06:11	that scholars are different than vectors and why we categorize
06:13	them differently . So the best example is gonna be
06:15	velocity , right ? I'm going 34 m per second
06:21	or I'm going negative 37 m per second . The
06:23	negative sign carries sign information . Now , that is
06:26	just we talked about so far , we just talked
06:28	about motion along one line where the only way we
06:31	can go is forward and backward , and the sign
06:33	takes care of it , right ? But And three
06:36	dimensional motion when I can go up and back and
06:39	and to and from you and so on . I
06:40	can go any direction in space . Then we'll have
06:43	vector . The vector is the total direction in three
06:47	dimensional space that you're that you're traveling . So you
06:49	may not be just going forward and backward , You
06:51	may be going this way that way , whatever .
06:53	And so we'll keep track of all of that .
06:55	You'll see how we do it later by keeping track
06:57	of the motion and X . Y and Z separately
07:00	. But as a quantity , when I throw the
07:01	baseball over there , in three dimensions , it has
07:05	a direction associated with it and a magnitude . So
07:07	we call it a vector quantity . Now another vector
07:10	quantity , one of the most important one is force
07:13	. When I push on something , I'm pushing in
07:15	a getting direction . Either I'm pushing along positive X
07:19	. Or I'll turn around and I'll push negative X
07:21	. Right ? So the force could have plus or
07:23	minus components also and that tells you which way you're
07:26	pushing because it obviously has a direction associated with pushing
07:29	. Another quantity is acceleration . We haven't talked about
07:35	acceleration yet , we're gonna define it and not too
07:37	long . But acceleration , you all know from a
07:40	car is when you speed up right ? Or you
07:43	might slow down . Actually . We call that deceleration
07:46	in everyday language when you slow down , deceleration in
07:49	physics , we don't really talk about deceleration . What
07:52	we say is you have acceleration in a positive sense
07:56	, meaning that you if you have positive acceleration you're
07:58	speeding up . But if you have a negative acceleration
08:02	, that's what we actually call deceleration in everyday language
08:04	is when you're slowing down . So you see the
08:06	acceleration variable has vector has a sign associated with a
08:11	positive acceleration means you're speeding up negative acceleration , not
08:14	something crazy . It just means you're slowing down so
08:17	it's giving you the direction of your speed up slow
08:20	down kind of thing . So that's why it's a
08:22	vector . There's a direction associated with acceleration , not
08:25	just the magnitude . And then we're gonna just list
08:27	the remaining here . We're going to get to these
08:30	much much later . But for instance , just kind
08:31	of give you a flavor magnetic field . Mhm .
08:37	Electric field . This this Mhm . I'll draw some
08:43	pictures in a second to help you visualize this .
08:44	But it doesn't matter so much for now because we
08:47	haven't studied this yet . But you've all seen magnets
08:49	, you know , magnets interact with each other ,
08:51	there's an invisible field . We call a magnetic field
08:53	. That field is a vector field . In other
08:55	words , every point in space around the magnet has
08:58	a strength but also direction . It's .2 different different
09:02	ways . Those are the field lines that come out
09:04	that we kind of can't see them , but that
09:05	we think they're there . We know they're there and
09:07	we call those vectors vector quantities at every point .
09:11	So this is the difference between scalar and vector .
09:13	So let's draw some quick examples here . So ,
09:17	if I have a ball um that I'm going to
09:21	throw and I'll throw it this direction , the velocity
09:25	Is 37 m/s . Notice that the the speed of
09:32	this ball is a is a magnitude . That's the
09:34	37 . So it has a magnitude check . And
09:38	it also has a direction . I'm going up into
09:40	the right , right . So it's a vector quantity
09:44	, that's all you have to ask . Does't have
09:46	a magnitude and a direction . If yes , it's
09:48	a vector , right ? If we were just talking
09:51	about speed we would tell you that it was going
09:53	37 m/s . But we wouldn't really tell you which
09:56	way it's going . So it wouldn't be a vector
09:58	. That's why speed is not a vector . All
10:00	right . Let's take another quick example . Let's say
10:02	we have a crate some kind of a box sitting
10:04	on the floor of my living room or whatever and
10:08	I'm going to push on it with a force so
10:10	I'm gonna represent that force as an arrow . So
10:13	force 10 Newtons . Now , I know we haven't
10:17	talked about the unit of force yet but just a
10:20	preview for you . It's going to be called a
10:22	Newton and we'll talk about why it's called the Newton
10:24	layers after Isaac Newton . But anyway , it's a
10:27	unit of force . I'm pushing with this magnitude but
10:30	I'm also pushing at this direction . So that's why
10:32	force is a vector . It has a size and
10:35	it also has a direction associated with it . Now
10:38	, just to give you a little bit more of
10:39	a flavor . Since we're going to be spending so
10:41	much time talking about vectors will just take just a
10:43	couple of seconds to draw a couple of other things
10:46	. Let's say you have a instead of a ball
10:49	, let's call it a proton . So I'm gonna
10:50	put a plus charge in the center . You all
10:53	probably taken basic chemistry or basic physics . So you
10:56	kind of know more or less what a proton is
10:58	. Those are the things in the center of the
11:00	nucleus . Well around this proton we say that this
11:04	electric field exist . And the way that we represented
11:07	our these arrows that kind of emanate kind of magically
11:11	. It's not really magic , but it comes and
11:13	emanates from the positive charge . And so this thing
11:17	, these lines here , the red lines . This
11:20	is called an electric field . Now , really you
11:26	can insert a word here . It's the electric vector
11:29	field . That's what it really . It's a field
11:30	of vectors . In other words , at every point
11:33	in space here , there is a value the length
11:36	of this era . We're going to get into that
11:38	actually , in just a second to kind of make
11:40	it more clear . But the length of the era
11:41	represents how strong the field is . And the direction
11:44	, of course , specifies which way it's pointing .
11:46	So there's this invisible field around all protons , right
11:50	? Or any kind of charged object that emanates from
11:52	it . And that electric field is what interacts with
11:55	other matter and pushes it all around . That's how
11:57	we are modern theories of electricity magnetism work and very
12:02	similar to that . You probably have an idea of
12:04	magnetic field , Okay , magnetic field , electric field
12:11	right now , how would we represent a magnetic field
12:14	? We'll just draw our friendly neighborhood bar magnet because
12:17	I know that you all have some experience with that
12:21	with playing around with those things and we can't see
12:24	the magnetic field , but we know it exists because
12:26	you know , we can interact and push on other
12:28	magnets and such . And so the way we talk
12:29	about it in physics is we say there's a invisible
12:33	magnetic field lines that come out here . This is
12:36	the kind of stuff you learn in Basic Basic Science
12:40	in like 3rd grade . But what you didn't learn
12:42	back then is that these lines have direction associated with
12:45	them . That's why we put the arrows . These
12:47	are like little arrows that exist all throughout space here
12:51	. Like this . So , you see how the
12:54	magnetic field has a strength and also a direction .
12:58	That's what the arrows are telling you . Which way
12:59	is it pointing ? And the electric field has the
13:01	strength in the direction and the force of the velocity
13:04	have a strength in their direction . That's why all
13:06	of them are are vector quantities . And then you
13:08	have these things that don't have any direction . All
13:10	of these examples , they don't have any direction .
13:12	So they're just called scalar quantities . Yeah . All
13:15	right . So how do we represent vectors ? I
13:18	told you way back here , I said for scale
13:20	, Urz , we use regular variables , just like
13:23	any variable . You just write them down . Okay
13:25	, vectors . You might see it written down slightly
13:28	different ways in different textbooks , but most of the
13:31	time we represent vectors . uh right with an arrow
13:41	on top . Sometimes you'll see it written slightly differently
13:45	, but most of the time this is what you'll
13:47	see if I I guess I should say the length
13:51	of the arrow . Well yeah , I'm kind of
13:56	getting maybe a little bit confusing length of arrow gives
14:05	you the magnitude in the direction of arrow , gives
14:12	you the direction , obviously . Yeah , I kinda
14:14	got a little ahead of myself with the way I
14:16	wrote this down , What I'm trying to say by
14:17	right with an arrow on the top is I'm trying
14:20	to say in an equation , like if I'm gonna
14:22	write down an equation for velocity or something like this
14:26	, then I'm going to use the variable V .
14:29	But I'm not gonna write it down like that because
14:31	if I leave it like that , you're going to
14:32	think it's a scaler , you're gonna think it's some
14:34	speed or something like this . So I write it
14:36	with a little arrow on top , then you know
14:38	that's a vector quantity . Okay , that's a velocity
14:43	right sector . If I'm gonna write down an equation
14:45	for acceleration , which we will have many , many
14:47	equations with acceleration , I'm gonna put a little arrow
14:49	on top , you're gonna know that that's the acceleration
14:52	, right ? If I'm gonna write down a force
14:55	, which maybe I'll write down Newton's second law of
14:57	motion or something like that . This is a force
14:58	vector little arrow on top . And just to give
15:03	you a crazy example that we're not gonna do any
15:04	time really soon . But if I have like e
15:06	with an arrow on top to go back to our
15:08	electric field , this would be an electric field vector
15:15	quantity . So anytime in an equation , when you
15:18	see a letter with an arrow on top , you
15:20	need to think that's a vector , right ? And
15:22	when you see an equation with a letter with nothing
15:24	at all , then it's a scaler quantity . Now
15:27	I am going to tell you that some books are
15:28	different . Some books instead of an arrow on top
15:31	, just put a bar on top . Okay .
15:34	The problem with that is that gets confusing because some
15:37	books indicate a bar on top , just a straight
15:40	bar being the average value of something . So like
15:43	the average of grades in the classroom might be great
15:45	variable with the bar on top . So it can
15:47	get confusing if you just put bars on top .
15:49	But when you're doing your homework , sometimes you just
15:51	quickly just put those bars . And so that's what
15:53	happens . Also , some books don't put the arrows
15:55	at all . They just bold . So you might
15:57	see V . But a capital V With a bold
16:00	, that means vector . So it depends on the
16:02	book you're using . But most of the time you're
16:04	gonna see vectors with an arrow on top . Now
16:07	this part of what I'm saying is dealing with what
16:11	I wrote down here right vectors with an arrow on
16:13	top . This is what I was trying to say
16:15	here . This stuff down here , the length of
16:17	the arrow in the direction of the arrow . This
16:19	is how we write down vectors . Um to actually
16:23	represent the values . So let's say for instance ,
16:25	to have a ball here And I'm gonna draw an
16:29	arrow like this . And I'm gonna say 10 m/s
16:33	and the arrow indicates the direction of throw with the
16:36	ball . And the magnitude is indicated by obviously the
16:41	number . Now let me go over here and let
16:43	me throw a different ball , but I'm gonna draw
16:45	it downward and I'm gonna represent this one is 15
16:49	m/s . Now it's important for you to see that
16:52	the lengths of these arrows are different . See this
16:54	one I'm representing , this length of an arrow is
16:56	10 and I'm representing this length of the arrow is
16:58	15 . So with vectors when you write them down
17:01	as arrows you draw . If it's a stronger value
17:04	, higher magnitude you draw the arrow longer . The
17:07	length of the arrow represents how big the number is
17:11	. The length of the arrow here is shorter ,
17:13	the length of the area here is bigger because obviously
17:14	these are different values . So just by looking at
17:17	a just by looking at a picture of arrows ,
17:22	you can see that the longer ones are stronger and
17:25	obviously the direction here is pointing down , the direction
17:27	here is pointing up , the directions are totally different
17:30	. Right ? So just to give you another example
17:32	, So this is velocity . Because we all have
17:35	experience with velocity . Let's quickly talk about force because
17:38	it won't be too long before we actually start dealing
17:40	with equations that deal with force . So just another
17:44	example . Um what if I have a vector this
17:47	long ? I'm representing that is 60 newtons . I
17:49	told you the unit of forces . Newton's , we'll
17:51	get to that later . And then I might have
17:53	another force here acting differently . That's going like this
17:56	. And this one might be 30 Nunes . Do
17:59	you see I'm not perfect with this . But do
18:00	you see how This vector is ? About half the
18:03	length maybe needs to be a little longer . About
18:05	half the length of this one because this 1 60
18:07	and this one is 30 . So it's the same
18:08	kind of thing , vectors in general , the length
18:11	of the era represents how how strong it is .
18:14	This one is shorter because it's a less force basically
18:18	. All right . Um , So reality is this
18:22	is what's going to happen . We're going to work
18:24	here in the first part by representing vectors as arrows
18:28	as these arrows because graphically it helps you visualize what
18:31	a vector is , the length of the arrows ,
18:33	how strong it is , the direction of the arrow
18:35	is , which way the thing is acting or moving
18:38	or whatever it is is doing whatever you're measuring .
18:40	Right ? So , we're gonna talk about arrows ,
18:41	we're gonna talk about graphical ways to write these vectors
18:44	down the next few sections . But then what's going
18:46	to actually happen is we're gonna throw away the graphical
18:48	pictures entirely . And you're not going to really use
18:50	those very much solving real physics problems . You will
18:54	use them some , but you're not gonna be drawing
18:56	tons of arrows all over the place to solve problems
18:59	, you're gonna write equations . So we use the
19:01	picture , the vector picture , the graphical cartoons to
19:03	visualize it . Then we're gonna gradually move into kind
19:06	of getting rid of that and and kind of not
19:08	needing that so much . It's very much like learning
19:10	to add negative numbers . We use the number line
19:13	first to show you how to add them , but
19:16	then after a while we kind of stopped using the
19:18	number line because you don't need it anymore . It
19:20	is a great tool start with . So that's what
19:21	we're gonna do here . So follow me on to
19:23	the next section , we're going to continue talking about
19:25	vectors and specifically representing them graphically , and how to
19:28	deal with vectors and physics .

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