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Scalars vs. Vectors
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About the lecture
In this mini-lecture, we discuss the difference between scalar quantities and vector quantities, focusing in particular on: (i) how scalar quantities describe magnitude (size); (ii) examples of scalars, such as time, distance, and mass; (iii) how vector quantities describe magnitude and direction; (iv) examples of vector quantities, such as velocity, force, weight, acceleration; (v) geometric vectors, which are vectors with a specified initial point and terminal point; (vi) a given vector’s length, which illustrates its magnitude, and the vector’s direction, given by the direction the arrow is pointing; (vii) an example of a vector describing a real-world displacement: a ship sailing northeast from point A to point B; (viii) two types of vector notation; (ix) two examples showing that vectors with the same magnitude and direction are the same vector, even if they are at different positions; and (x) the zero vector, which has no length and no direction.
About the lecturer
Emille Lawrence is a Professor in the Department of Mathematics and Statistics at the University of San Francisco. She is a low-dimensional topologist and her mathematical interests include braid groups, geometric group theory, and spatial graphs. She is passionate about broadening participation in the mathematical sciences through outreach and mentoring. In 2021, Professor Lawrence received the Association for Women in Mathematics (AWM) Service Award for her work to increase diversity in those participating in the AWM.
Hi.
00:00:05My name is Emil Lawrence and I'm a professor at the
00:00:06University of San Francisco and the Mathematics and Statistics Department.
00:00:09This is a course vectors one giving an introduction to vectors.
00:00:14How to assign coordinates two vectors doing arithmetic with vectors
00:00:18and standard unit vectors in R two and r three.
00:00:23So we'll start today by talking about the
00:00:27difference between scalar quantities and vector quantities.
00:00:31So a scalar quantity answers the question. How much?
00:00:35Okay, so examples of scalar quantities, um, would be time, distance,
00:00:54mass
00:01:04and et cetera.
00:01:06So, in contrast, a vector quantity
00:01:08not only answers the question, how much, but in what direction
00:01:13which direction?
00:01:29Examples of vector quantities include Velocity,
00:01:31force,
00:01:38wait
00:01:41acceleration,
00:01:43and the list goes on.
00:01:46So this lesson is about vectors.
00:01:51So we're going to focus mainly on vector quantities now because
00:01:53vector quantities have a direction and what we call a magnitude,
00:01:59how much we often use geometric tools to describe vectors.
00:02:04So what is a geometric vector?
00:02:09So here's an example of a geometric factor,
00:02:14just
00:02:20an arrow
00:02:21written in space.
00:02:22It has a start, which I'm calling a
00:02:24and a finish, which I'm calling B. We call a the initial point of vector
00:02:27and be the terminal point of the vector.
00:02:38And the length of the vector answers the question for us, how much
00:02:50the direction of the vector or the direction in which the arrow is pointing.
00:03:10That's just giving us the direction in which we're going.
00:03:15Okay, so this is a good description, A geometric description of Vector.
00:03:19Um, and this vector could be describing displacement.
00:03:25Let's just say we have maybe a ship.
00:03:36The shift started at point A.
00:03:39The ship sailed 100 miles northeast and set up dot at point B.
00:03:42So this is an example of a vector, and maybe it could be a displacement,
00:03:49So we have some common notation for vectors.
00:03:54So the vector that I've drawn here,
00:03:59I'm going to use a capital and be with an arrow overtop.
00:04:01Another common notation would be maybe just using a lower case,
00:04:06letter V and sometimes in textbooks.
00:04:11You see this in bold.
00:04:14I can't draw and bold on my iPad here, so I'm just going to put a line over top of the V.
00:04:15So these are both very common notations for vectors, So
00:04:21Okay, so let's go back to our ship. Example. What if we had another ship?
00:04:27So here's our first ship
00:04:44that sailed from point A to point B.
00:04:48Maybe it's represented by this vector V.
00:04:51But what if we had another ship
00:04:54maybe
00:04:5650 miles away? But it also sailed
00:04:57in the same direction
00:05:01northeast
00:05:03for 100 miles.
00:05:04But it didn't start and end at the same point.
00:05:06It started at point C
00:05:08and ended at point D.
00:05:10Do you think these two vectors
00:05:15V and W
00:05:17should be
00:05:20the same?
00:05:21Well,
00:05:23they have the same length
00:05:24and they have the same direction, so they absolutely should be the same vectors. So
00:05:25this is our equivalence
00:05:31for vectors.
00:05:33If two vectors have the same length
00:05:35in the same direction, we say they're equal.
00:05:38They don't have to be sitting in the same place in space.
00:05:47So in fact,
00:05:57all of the vectors that I'm drawing here trying to make
00:06:04these look like the same direction in the same length,
00:06:08all of these vectors we call the same.
00:06:11Okay,
00:06:14okay.
00:06:39Also, I want to know there's a special vector called the zero Vector.
00:06:41The zero vector has no length
00:06:51and no particular direction, so I actually can't even draw the zero vector.
00:06:58Maybe if I were to draw something,
00:07:04I would draw the zero Vector is just a point,
00:07:09and I'll note
00:07:12that that's the zero vector by just writing zero with the bar over it.
00:07:15So in this video, we've given a brief overview of what a vector quantity is.
00:07:19We've also described vectors geometrically using an arrow talking
00:07:26about the initial point in the Terminal Point.
00:07:32And we've also defined the zero Vector and the next video.
00:07:35We'll talk about how to do arithmetic with vectors.
00:07:40
Cite this Lecture
APA style
Lawrence, E. (2022, August 30). 1. Measurements and Uncertainties - Scalars vs. Vectors [Video]. MASSOLIT. https://massolit.io/options/1-measurements-and-uncertainties?auth=0&lesson=8660&option=14516&type=lesson
MLA style
Lawrence, E. "1. Measurements and Uncertainties – Scalars vs. Vectors." MASSOLIT, uploaded by MASSOLIT, 30 Aug 2022, https://massolit.io/options/1-measurements-and-uncertainties?auth=0&lesson=8660&option=14516&type=lesson