Timeline for What is the rigorous and formal definition for the direction pointed by a gradient?
Current License: CC BY-SA 4.0
5 events
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| Jan 12, 2022 at 9:51 | comment | added | nbro | You could also view the direction as the angle that the vector makes with the axes. In that way, you might think that you don't need to define it with respect to a unit vector, but you would still need to define it with respect to a basis vector, so a vector. I'm not sure if there's any other "better" way to represent the direction of a vector. | |
| Jan 12, 2022 at 9:50 | comment | added | nbro | @hanugm A vector is defined by a magnitude and direction. So, a vector cannot just be a line (i.e. without a direction) or a direction (i.e. without a magnitude). The definition may look circular indeed, because you use a vector to define the direction of another vector, but we can view this unit vector as representing the directions of infinitely many other vectors (if $u$ is a unit vector, then $c u$ all have the same direction, for all $c \in \mathbb{R}$), so this unit vector is not specific to the original vector you want to find the direction of. | |
| Nov 16, 2021 at 22:30 | comment | added | hanugm | I think the definition for direction is becoming circular here. So, the direction pointed by a vector is the same as the direction pointed by the unit vector of that vector. But, what is meant by "direction pointed by a vector" mathematically? | |
| Nov 16, 2021 at 21:19 | comment | added | nbro | So, it may be a good idea to emphasize 2 things. 1. The gradient vector is not special. If a vector of $n$ real numbers, then it's a vector in $n$-d space and you cannot really visualize it if $n > 3$. 2. $\frac{u}{\|u\|}$ is the unit vector that has the same direction as $u$. Why do you compute the unit vector? Because you can have vectors of different magnitude that point in the same direction, and they are not the same vector because they have a different magnitude: e.g, in physics, you can have 2 vectors that represent 2 forces in the same direction but different magnitude. | |
| Nov 16, 2021 at 19:38 | history | answered | Taw | CC BY-SA 4.0 |