IB · Mathematics Analysis and Approaches (AA)
AAHL 3.14.4 Vectors and kinematics
Learn AAHL 3.14.4 Vectors and kinematics in this free IB Mathematics Analysis and Approaches (AA) video lesson for AHL 3.14—Vector equation of line.
Learn AAHL 3.14.4 Vectors and kinematics in this free IB Mathematics Analysis and Approaches (AA) video lesson for AHL 3.14—Vector equation of line.
Learn AAHL 3.14.4 Vectors and kinematics in this free IB Mathematics Analysis and Approaches (AA) video lesson for AHL 3.14—Vector equation of line.
The video discusses the application of vectors in kinematics, using examples of an airplane and two ships to illustrate the concept. The position of the airplane is modeled with the vector equation , where represents time in seconds.
For the ships, Ship A's position is given by , while Ship B starts at and moves with a velocity of . The distance between the ships after two hours is calculated using the distance formula, resulting in a distance of kilometers.
00:00Hi guys, so here we're
00:03going to look at a
00:03nice application of vectors and
00:05that is in kinematics. So
00:09let me just introduce you
00:11to this using this beautiful
00:13picture of an airplane that
00:14I've drawn. We can model
00:19the position of an airplane
00:24or a boat or anything
00:25moving in straight lines using
00:28vector equations. So for example,
00:33this plane has position vector
00:38750 plus t times minus
00:43time, now that could be
00:44in hours or seconds, I
00:47think in this particular case,
00:49we're going to assume it's
00:51seconds. Now, what's happening is
00:54the plane
00:56the plane has taken off.
00:59So let's say this is
01:00the ground here. So his
01:04position vector is 7, 5,
01:070. So this point is
01:09like 7, 5, 0. Note
01:110 because that is, well,
01:15in this particular example, the
01:16z axis is our kind
01:20of vertical axis in which
01:22the plane
01:24takes off and 7 .5.
01:28This is like your x
01:30-coordinate and y -coordinate on
01:31the ground, but zero or
01:33the z -coordinate is how
01:36far up in the air
01:37he is. So after one
01:39second he is three meters
01:41in the air. After 10
01:43seconds he's 30 meters in
01:44the air and these two
01:46describe ways and the x
01:48-y plane. So what happens
01:51is every
01:52Three second he travels this
01:56much negative five in the
01:58x direction, four in the
02:00y direction, and three in
02:01the z direction. So he's
02:03traveling this much, this vector
02:06per second. So every second
02:08we're going to add this
02:09on. So think about what
02:11does that mean when you
02:12travel at a certain amount
02:14or when your displacement is
02:16something per second? It means
02:19it's this is your
02:20velocity. So the direction vector,
02:22when we write a vector
02:25equation of a line like
02:28this where the parameter is
02:30actually the time, this is
02:32the velocity. And then the
02:35magnitude of this vector, the
02:38magnitude of the direction vector,
02:44which is, it gives us
02:47the speed.
02:48The speed, remember the speed
02:50is a scalar, so we
02:52don't need the direction, so
02:53the speed is just five
02:55squared plus four squared plus
02:57three squared, which is square
03:01root of 25 plus 16
03:04is 41 plus nine, which
03:09is 50, exactly, square root
03:11of 50. So he's going
03:14at a speed of,
03:16square root of 50 meters
03:18per second. So I don't
03:20know if that's fast for
03:21an airplane or not. Let
03:22me think that would be
03:23approximately seven meters per second.
03:27That's probably not that fast
03:29for an airplane, but whatever.
03:31It's just an example. So
03:32that's what it means. That's
03:33kind of his starting point.
03:35That's his velocity. And then
03:38the whole equation gives us
03:40his position after any amount
03:42of time. So if I
03:43said to you, well, what's
03:44the plane
03:44position after, I don't know,
03:49a hundred seconds, you'd sub
03:50in a hundred there and
03:52you get his position vector.
03:55This obviously will model the
03:57plane from when he takes
03:58off until he reaches his
04:00kind of maximum altitude or
04:03whatever and then obviously it
04:05changes this. The Z component
04:12would be zero again because
04:14he's just gonna maintain that
04:16altitude. Okay, that's a plane
04:20in 3D. This example here,
04:22I'm actually gonna look at
04:23boats, which is more in
04:24two dimensions. So it says,
04:27ship A leaves a port
04:28located at the origin and
04:30moves with a velocity 3i
04:32plus 5j. Guys, whenever I
04:34have a question, whenever I
04:38have a question, I like
04:40to whatever question. I like
04:43to draw a diagram just
04:45to see what's actually happening.
04:47Let me get rid of
04:48this. I want to do
04:51this in black. Imagine I
04:59have... here's my... whatever it
05:02is my plane, my set
05:05of axes, and this ship
05:08is leaving the
05:08and he's moving with a
05:10velocity of three i plus
05:11five j. So let's say
05:12that's three i plus five
05:17j, something like this. He's
05:19moving like this. Fine. That
05:23direction. And then ship b
05:25starts 18 kilometers north. So
05:29he is moving while he's
05:33starting up here somewhere. 18
05:35kilometers north. So directly
05:36north and he's moving it
05:394i minus 2j so let's
05:41go 4 minus 2, 4
05:48minus 2 something like that.
05:53Okay so he's going that
05:54direction, he's going that direction.
05:56It says find the distance
05:57between the ships after two
06:00hours. Okay so
06:04Shabay leaves the port and
06:06he's moving a velocity of
06:083i plus 5j. So ship
06:10A, we can model ship
06:12A's path using this vector
06:15equation, or equals position vector
06:180, 0, he started at
06:200, 0 plus t times
06:24because we're looking at its
06:31kilometers per hour.
06:33Every hour he moves, every
06:36hour he is going to
06:38move 3i plus 5j, so
06:42this becomes 3 3 5.
06:46So that's ship A's position
06:50vector after a given time.
06:53And ship B's position after
06:56a given time will be
06:58or equal. So he starts
07:00at z
07:01zero 18 because he's zero
07:05along the x -axis and
07:0718 up here plus t
07:10times and his velocity is
07:11four minus two, four minus
07:14two like this. So the
07:18question says find the distance
07:19between the ships after two
07:21hours. So after two hours,
07:25after two hours, ship A.
07:29A, what is ship A's
07:33position vector? Well, ship A's
07:35position vector is 0, 0
07:38plus 2 times because it's
07:41now 2 hours, so T
07:42is 2. 2 times 3,
07:445, which is just 6,
07:4910. So his position vector
07:50is 6, 10. And ship
07:53B, ship
07:57B's position vector is going
07:59to be 0 .18 plus
08:032 times 4 negative 2,
08:07which gives us 8, 8,
08:10and then 18 minus 4
08:14is 14, 8, 14. That's
08:18his position vector. That's his
08:20position vector. Let me draw
08:21another little graph.
08:25just to show you what's
08:26happening here. So he is
08:28at this guy's at 6
08:3110. So let's say 6
08:3410. Something like this. That's
08:38his point is 6 10
08:41and ship B. His point
08:44is 8, 8, 14. So
08:48he's here at 8, 14.
08:50So now we just have
08:51those are the points we
08:52now
08:53just have to find the
08:54distance between two points, which
08:58is, we know this formula.
09:00Distance is equal to the
09:04square root of, and it's
09:06this minus the squared plus
09:08this minus the squared. So
09:09it's eight minus six squared
09:11plus 14 minus 10 squared.
09:16This is in the formula.
09:17But look, guys, if you
09:18don't know what I'm doing,
09:20but hopefully, you know, the
09:21distance
09:21since between two points formula
09:22by now. So this is
09:23going to be four, let's
09:29two squared plus four plus
09:33four squared, which is 16,
09:38giving us root 20 kilometers
09:42exactly. Okay, that's it, that's
09:47my lesson on vectors with
09:49kinematics. Obviously the main things
09:52to note is we can
09:54use the vector of equation
09:55of a line to model
09:57the path of some object,
10:03be it a plane or
10:04a ship or whatever it
10:05might be. We can do
10:06it in three dimensions and
10:08two dimensions. The position vector
10:10is kind of its starting
10:12point because it's like when
10:14t is zero. The initial
10:16is initial position.
10:17And then because I'm adding
10:20the direction vector, or I'm
10:22adding a direction vector for
10:23every given time, whether it's
10:25a second or a kilometer,
10:27that is by definition the
10:29velocity and the magnitude of
10:31the velocity gives us the
10:33speed. That's it. Hope that
10:36all made sense and I'll
10:38see you in the next
10:38lesson.