Math 21a Summer 2023
Multivariable Calculus
Here is what chat GPT produced during class on Tuesday to the question
"What is a vector":
A vector is a mathematical entity that has both a magnitude (size
or length) and direction. It's often used in fields like physics,
engineering, and computer science to represent things like forces,
velocities, or even positions in space.
Vectors can be represented in a few different ways:
Graphically: A vector can be represented as an arrow in space. The length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector.
Numerically: A vector can also be represented as an ordered list of numbers, like (2, 3) or (5, 1, 2). Each of these numbers corresponds to a component of the vector in a particular direction. For example, in two dimensions, the numbers correspond to the x and y components of the vector. In three dimensions, there would be a third number for the z component.
It's important to note that two vectors are equal if they have the same magnitude and direction, regardless of their initial position. In other words, if you take a vector and move it around without changing its length or the direction it's pointing, it's still the same vector. Also, vectors can be added, subtracted, and multiplied in certain ways that are different from the ways we usually add, subtract, and multiply numbers. These operations are defined in such a way as to make them useful in a wide variety of scientific and engineering contexts. For example, adding two vectors results in a new vector that is obtained by placing the initial point of the second vector at the terminal point of the first vector, and then drawing a vector from the initial point of the first vector to the terminal point of the second vector.
Comment: It is common fallacy to define a vector as a quantity with magnitude and direction. There are various problems with this ``definition". First of all, it defines something using two other non-defined quantities, namely "magnitude" and "direction". But "direction" requires other concepts like "vector" or "angle" so that the definition is circular. More serious is that the definition does not work for the zero vector. The zero vector has no direction. Excluding the zero vector is serious as 0 is an essential part of a vector space. Third, the definition is so generic that it applies to various other things. According to that definition, a movie would be a vector as a movie has a magnitude (length) and direction (like a director). Imprecise definitions are a common problem to science because they lead to misunderstandings or wrong conclusions.
The following picture was AI generated. (In my view, one of the most important rules we have to follow is to acknowledge the use of AI, whenever it is used. One reason is copy right issues. AI generated content is not copyright protected and this should be so. If I had drawn the picture below myself, then it would be me, the artist who has created it. Now, it was me who asked AI (the artist) to generate a picture of a vector in a Caribean setting with nice clear water. Of course, I had to use some imagination to trigger drawing the picture but I have not drawn the picture. I was surprised by the outcome. Click on the picture to see it larger.
Graphically: A vector can be represented as an arrow in space. The length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector.
Numerically: A vector can also be represented as an ordered list of numbers, like (2, 3) or (5, 1, 2). Each of these numbers corresponds to a component of the vector in a particular direction. For example, in two dimensions, the numbers correspond to the x and y components of the vector. In three dimensions, there would be a third number for the z component.
It's important to note that two vectors are equal if they have the same magnitude and direction, regardless of their initial position. In other words, if you take a vector and move it around without changing its length or the direction it's pointing, it's still the same vector. Also, vectors can be added, subtracted, and multiplied in certain ways that are different from the ways we usually add, subtract, and multiply numbers. These operations are defined in such a way as to make them useful in a wide variety of scientific and engineering contexts. For example, adding two vectors results in a new vector that is obtained by placing the initial point of the second vector at the terminal point of the first vector, and then drawing a vector from the initial point of the first vector to the terminal point of the second vector.
Comment: It is common fallacy to define a vector as a quantity with magnitude and direction. There are various problems with this ``definition". First of all, it defines something using two other non-defined quantities, namely "magnitude" and "direction". But "direction" requires other concepts like "vector" or "angle" so that the definition is circular. More serious is that the definition does not work for the zero vector. The zero vector has no direction. Excluding the zero vector is serious as 0 is an essential part of a vector space. Third, the definition is so generic that it applies to various other things. According to that definition, a movie would be a vector as a movie has a magnitude (length) and direction (like a director). Imprecise definitions are a common problem to science because they lead to misunderstandings or wrong conclusions.
The following picture was AI generated. (In my view, one of the most important rules we have to follow is to acknowledge the use of AI, whenever it is used. One reason is copy right issues. AI generated content is not copyright protected and this should be so. If I had drawn the picture below myself, then it would be me, the artist who has created it. Now, it was me who asked AI (the artist) to generate a picture of a vector in a Caribean setting with nice clear water. Of course, I had to use some imagination to trigger drawing the picture but I have not drawn the picture. I was surprised by the outcome. Click on the picture to see it larger.
