[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.63,0:00:02.09,Default,,0000,0000,0000,,- [Instructor] In other\Nvideos, we have talked
Dialogue: 0,0:00:02.09,0:00:04.10,Default,,0000,0000,0000,,about how a vector can\Nbe completely defined
Dialogue: 0,0:00:04.10,0:00:06.88,Default,,0000,0000,0000,,by a magnitude and a\Ndirection, you need both.
Dialogue: 0,0:00:06.88,0:00:08.26,Default,,0000,0000,0000,,And here we have done that.
Dialogue: 0,0:00:08.26,0:00:09.88,Default,,0000,0000,0000,,We have said that the magnitude
Dialogue: 0,0:00:09.88,0:00:12.57,Default,,0000,0000,0000,,of vector a is equal to three units,
Dialogue: 0,0:00:12.57,0:00:15.19,Default,,0000,0000,0000,,these parallel lines here on both sides,
Dialogue: 0,0:00:15.19,0:00:17.17,Default,,0000,0000,0000,,it looks like a double absolute value.
Dialogue: 0,0:00:17.17,0:00:19.09,Default,,0000,0000,0000,,That means the magnitude of vector a.
Dialogue: 0,0:00:19.09,0:00:23.15,Default,,0000,0000,0000,,And you can also specify\Nthat visually by making sure
Dialogue: 0,0:00:23.15,0:00:26.20,Default,,0000,0000,0000,,that the length of this vector\Narrow is three units long.
Dialogue: 0,0:00:26.20,0:00:27.56,Default,,0000,0000,0000,,And we also have its direction.
Dialogue: 0,0:00:27.56,0:00:29.61,Default,,0000,0000,0000,,We see the direction of\Nvector a is 30 degrees
Dialogue: 0,0:00:29.61,0:00:32.27,Default,,0000,0000,0000,,counter-clockwise of due East.
Dialogue: 0,0:00:32.27,0:00:34.86,Default,,0000,0000,0000,,Now in this video, we're\Ngonna talk about other ways
Dialogue: 0,0:00:34.86,0:00:38.22,Default,,0000,0000,0000,,or another way to specify\Nor to define a vector.
Dialogue: 0,0:00:38.22,0:00:41.05,Default,,0000,0000,0000,,And that's by using components.
Dialogue: 0,0:00:41.05,0:00:42.53,Default,,0000,0000,0000,,And the way that we're gonna do it is,
Dialogue: 0,0:00:42.53,0:00:44.10,Default,,0000,0000,0000,,we're gonna think about the tail
Dialogue: 0,0:00:44.10,0:00:47.30,Default,,0000,0000,0000,,of this vector and the\Nhead of this vector.
Dialogue: 0,0:00:47.30,0:00:50.45,Default,,0000,0000,0000,,And think about as we go\Nfrom the tail to the head,
Dialogue: 0,0:00:50.45,0:00:53.99,Default,,0000,0000,0000,,what is our change in x?
Dialogue: 0,0:00:53.99,0:00:55.18,Default,,0000,0000,0000,,And we could see our change
Dialogue: 0,0:00:55.18,0:00:58.34,Default,,0000,0000,0000,,in x would be that right over there.
Dialogue: 0,0:00:58.34,0:01:00.98,Default,,0000,0000,0000,,We're going from this x\Nvalue to this x value.
Dialogue: 0,0:01:00.98,0:01:05.37,Default,,0000,0000,0000,,And then what is going\Nto be our change in y.
Dialogue: 0,0:01:05.37,0:01:07.98,Default,,0000,0000,0000,,And if we're going from\Ndown here to up here,
Dialogue: 0,0:01:07.98,0:01:12.31,Default,,0000,0000,0000,,our change in y, we can\Nalso specify like that.
Dialogue: 0,0:01:12.31,0:01:13.50,Default,,0000,0000,0000,,So let me label these.
Dialogue: 0,0:01:13.50,0:01:18.50,Default,,0000,0000,0000,,This is my change in x, and\Nthen this is my change in y.
Dialogue: 0,0:01:19.06,0:01:19.92,Default,,0000,0000,0000,,And if you think about it,
Dialogue: 0,0:01:19.92,0:01:22.78,Default,,0000,0000,0000,,if someone told you your\Nchange in x and change in y,
Dialogue: 0,0:01:22.78,0:01:25.39,Default,,0000,0000,0000,,you could reconstruct this\Nvector right over here
Dialogue: 0,0:01:25.39,0:01:27.49,Default,,0000,0000,0000,,by starting here, having that change in x,
Dialogue: 0,0:01:27.49,0:01:31.20,Default,,0000,0000,0000,,then having the change in y\Nand then defining where the tip
Dialogue: 0,0:01:31.20,0:01:34.74,Default,,0000,0000,0000,,of the vector would be\Nrelative to the tail.
Dialogue: 0,0:01:34.74,0:01:38.80,Default,,0000,0000,0000,,The notation for this is\Nwe would say that vector a
Dialogue: 0,0:01:38.80,0:01:42.87,Default,,0000,0000,0000,,is equal to, and we'll have parenthesis,
Dialogue: 0,0:01:42.87,0:01:46.29,Default,,0000,0000,0000,,and we'll have our change\Nin x comma, change in y.
Dialogue: 0,0:01:46.29,0:01:47.78,Default,,0000,0000,0000,,And so if we wanted to get tangible
Dialogue: 0,0:01:47.78,0:01:50.34,Default,,0000,0000,0000,,for this particular\Nvector right over here,
Dialogue: 0,0:01:50.34,0:01:53.55,Default,,0000,0000,0000,,we know the length of\Nthis vector is three.
Dialogue: 0,0:01:53.55,0:01:55.54,Default,,0000,0000,0000,,Its magnitude is three.
Dialogue: 0,0:01:55.54,0:01:58.35,Default,,0000,0000,0000,,We know that this is, since\Nthis is going due horizontally
Dialogue: 0,0:01:58.35,0:02:00.29,Default,,0000,0000,0000,,and then this is going\Nstraight up and down.
Dialogue: 0,0:02:00.29,0:02:02.42,Default,,0000,0000,0000,,This is a right triangle.
Dialogue: 0,0:02:02.42,0:02:05.17,Default,,0000,0000,0000,,And so we can use a little\Nbit of geometry from the past.
Dialogue: 0,0:02:05.17,0:02:08.02,Default,,0000,0000,0000,,Don't worry if you need a little\Nbit of a refresher on this,
Dialogue: 0,0:02:08.02,0:02:09.62,Default,,0000,0000,0000,,but we could use a little bit of geometry,
Dialogue: 0,0:02:09.62,0:02:11.49,Default,,0000,0000,0000,,or a little bit of\Ntrigonometry to establish,
Dialogue: 0,0:02:11.49,0:02:13.61,Default,,0000,0000,0000,,if we know this angle,\Nif we know the length
Dialogue: 0,0:02:13.61,0:02:17.21,Default,,0000,0000,0000,,of this hypotenuse, that\Nthis side that's opposite
Dialogue: 0,0:02:17.21,0:02:20.18,Default,,0000,0000,0000,,the 30 degree angle is gonna\Nbe half the hypotenuse,
Dialogue: 0,0:02:20.18,0:02:22.02,Default,,0000,0000,0000,,so it's going to be 3/2.
Dialogue: 0,0:02:22.02,0:02:24.20,Default,,0000,0000,0000,,And that the change in x is going to be
Dialogue: 0,0:02:24.20,0:02:26.96,Default,,0000,0000,0000,,the square root of three times the 3/2.
Dialogue: 0,0:02:26.96,0:02:31.08,Default,,0000,0000,0000,,So it's going to be three,\Nsquare roots of three over two.
Dialogue: 0,0:02:31.08,0:02:33.98,Default,,0000,0000,0000,,And so up here, we would\Nwrite our x component
Dialogue: 0,0:02:33.98,0:02:37.68,Default,,0000,0000,0000,,is three times the square\Nroot of three over two.
Dialogue: 0,0:02:37.68,0:02:42.42,Default,,0000,0000,0000,,And we would write that\Nthe y component is 3/2.
Dialogue: 0,0:02:42.42,0:02:43.82,Default,,0000,0000,0000,,Now I know a lot of you might be thinking
Dialogue: 0,0:02:43.82,0:02:47.26,Default,,0000,0000,0000,,this looks a lot like coordinates\Nin the coordinate plane,
Dialogue: 0,0:02:47.26,0:02:48.58,Default,,0000,0000,0000,,where this would be the x coordinate
Dialogue: 0,0:02:48.58,0:02:50.30,Default,,0000,0000,0000,,and this would be the y coordinate.
Dialogue: 0,0:02:50.30,0:02:51.97,Default,,0000,0000,0000,,But when you're dealing with vectors,
Dialogue: 0,0:02:51.97,0:02:54.61,Default,,0000,0000,0000,,that's not exactly the interpretation.
Dialogue: 0,0:02:54.61,0:02:57.00,Default,,0000,0000,0000,,It is the case that if the vector's tail
Dialogue: 0,0:02:57.00,0:03:00.86,Default,,0000,0000,0000,,were at the origin right\Nover here, then its head
Dialogue: 0,0:03:00.86,0:03:04.67,Default,,0000,0000,0000,,would be at these coordinates\Non the coordinate plane.
Dialogue: 0,0:03:04.67,0:03:07.47,Default,,0000,0000,0000,,But we know that a vector is not defined
Dialogue: 0,0:03:07.47,0:03:10.18,Default,,0000,0000,0000,,by its position, by the\Nposition of the tail.
Dialogue: 0,0:03:10.18,0:03:12.20,Default,,0000,0000,0000,,I could shift this vector around wherever
Dialogue: 0,0:03:12.20,0:03:13.84,Default,,0000,0000,0000,,and it would still be the same vector.
Dialogue: 0,0:03:13.84,0:03:15.59,Default,,0000,0000,0000,,It can start wherever.
Dialogue: 0,0:03:15.59,0:03:19.00,Default,,0000,0000,0000,,So when you use this\Nnotation in a vector context,
Dialogue: 0,0:03:19.00,0:03:21.44,Default,,0000,0000,0000,,these aren't x coordinates\Nand y coordinates.
Dialogue: 0,0:03:21.44,0:03:26.44,Default,,0000,0000,0000,,This is our change in x,\Nand this is our change in y.
Dialogue: 0,0:03:27.07,0:03:28.48,Default,,0000,0000,0000,,Let me do one more example to show
Dialogue: 0,0:03:28.48,0:03:30.88,Default,,0000,0000,0000,,that we can actually go the other way.
Dialogue: 0,0:03:30.88,0:03:34.79,Default,,0000,0000,0000,,So let's say I defined some vector b,
Dialogue: 0,0:03:34.79,0:03:39.20,Default,,0000,0000,0000,,and let's say that its x\Ncomponent is square root of two.
Dialogue: 0,0:03:39.20,0:03:43.52,Default,,0000,0000,0000,,And let's say that its y\Ncomponent is square root of two.
Dialogue: 0,0:03:43.52,0:03:46.26,Default,,0000,0000,0000,,So let's think about what\Nthat vector would look like.
Dialogue: 0,0:03:46.26,0:03:49.38,Default,,0000,0000,0000,,So it would, if this is its tail,
Dialogue: 0,0:03:49.38,0:03:51.41,Default,,0000,0000,0000,,and its x component which is its change
Dialogue: 0,0:03:51.41,0:03:53.03,Default,,0000,0000,0000,,in x is square root of two.
Dialogue: 0,0:03:53.03,0:03:55.46,Default,,0000,0000,0000,,So it might look something like this.
Dialogue: 0,0:03:55.46,0:04:00.46,Default,,0000,0000,0000,,So that would be change in x\Nis equal to square root of two.
Dialogue: 0,0:04:00.80,0:04:03.98,Default,,0000,0000,0000,,And then its y component would\Nalso be square root of two.
Dialogue: 0,0:04:03.98,0:04:07.23,Default,,0000,0000,0000,,So I could write our change in y over here
Dialogue: 0,0:04:07.23,0:04:08.97,Default,,0000,0000,0000,,is square root of two.
Dialogue: 0,0:04:08.97,0:04:12.85,Default,,0000,0000,0000,,And so the vector would\Nlook something like this.
Dialogue: 0,0:04:12.85,0:04:17.85,Default,,0000,0000,0000,,It would start here and\Nthen it would go over here,
Dialogue: 0,0:04:18.58,0:04:20.59,Default,,0000,0000,0000,,and we can use a little bit of geometry
Dialogue: 0,0:04:20.59,0:04:21.98,Default,,0000,0000,0000,,to figure out the magnitude
Dialogue: 0,0:04:21.98,0:04:24.26,Default,,0000,0000,0000,,and the direction of this vector.
Dialogue: 0,0:04:24.26,0:04:26.76,Default,,0000,0000,0000,,You can use the Pythagorean\Ntheorem to establish
Dialogue: 0,0:04:26.76,0:04:28.76,Default,,0000,0000,0000,,that this squared plus this squared
Dialogue: 0,0:04:28.76,0:04:30.41,Default,,0000,0000,0000,,is gonna be equal to that squared.
Dialogue: 0,0:04:30.41,0:04:32.38,Default,,0000,0000,0000,,And if you do that,\Nyou're going to get this
Dialogue: 0,0:04:32.38,0:04:34.51,Default,,0000,0000,0000,,having a length of two, which tells you
Dialogue: 0,0:04:34.51,0:04:39.37,Default,,0000,0000,0000,,that the magnitude of\Nvector b is equal to two.
Dialogue: 0,0:04:39.37,0:04:42.42,Default,,0000,0000,0000,,And if you wanted to figure\Nout this angle right over here,
Dialogue: 0,0:04:42.42,0:04:43.87,Default,,0000,0000,0000,,you could do a little bit of trigonometry
Dialogue: 0,0:04:43.87,0:04:46.11,Default,,0000,0000,0000,,or even a little bit\Nof geometry recognizing
Dialogue: 0,0:04:46.11,0:04:49.50,Default,,0000,0000,0000,,that this is going to be a\Nright angle right over here,
Dialogue: 0,0:04:49.50,0:04:52.13,Default,,0000,0000,0000,,and that this side and that\Nside have the same length.
Dialogue: 0,0:04:52.13,0:04:53.41,Default,,0000,0000,0000,,So these are gonna be the same angles
Dialogue: 0,0:04:53.41,0:04:55.60,Default,,0000,0000,0000,,which are gonna be 45 degree angles.
Dialogue: 0,0:04:55.60,0:04:58.69,Default,,0000,0000,0000,,And so just like that, you could\Nalso specify the direction,
Dialogue: 0,0:04:58.69,0:05:02.77,Default,,0000,0000,0000,,45 degrees counter-clockwise of due East.
Dialogue: 0,0:05:02.77,0:05:05.36,Default,,0000,0000,0000,,So hopefully you appreciate\Nthat these are equivalent ways
Dialogue: 0,0:05:05.36,0:05:06.54,Default,,0000,0000,0000,,of representing a vector.
Dialogue: 0,0:05:06.54,0:05:08.95,Default,,0000,0000,0000,,You either can have a\Nmagnitude and a direction,
Dialogue: 0,0:05:08.95,0:05:10.20,Default,,0000,0000,0000,,or you can have your components
Dialogue: 0,0:05:10.20,0:05:12.35,Default,,0000,0000,0000,,and you can go back and\Nforth between the two.
Dialogue: 0,0:05:12.35,0:05:15.28,Default,,0000,0000,0000,,And we'll get more practice\Nof that in future videos.