Identifying Surfaces for Vectors: k, l, m, n, \hat{u}

In summary, the surfaces in this problem all involve a fixed unit vector, with k, l, m, and n as fixed values. For a), the surface is a sphere with radius k. For b), the surface is a cone with a constant angle between the r vector and u hat. In c), the cosine of the angle between r and u equals m, which results in a cone with a variable angle between 0 and pi. And for d), the surface is a line defined by r minus the projection of r in the u hat direction.
  • #1
thenewbosco
187
0
The question reads:
Identify the following surfaces given that k, l, m, n are fixed values and [tex]\hat{u}[/tex] is a fixed unit vector.

a) [tex]|\overrightarrow{r}|=k[/tex]
b) [tex] \hat{r}\cdot \hat u=l[/tex]
c) [tex] \overrightarrow{r} \cdot \hat{u} = m|\overrightarrow{r}|[/tex] for [tex] -1 \leq 1[/tex]
d)[tex]|\overrightarrow{r} - (\overrightarrow{r}\cdot\hat{u})\hat{u}|=n[/tex]

I am to consider the both the variability in magnitude and direction of [tex]\overrightarrow{r}[/tex]

i was just wondering if the following are correct.
for a) it seems pretty obvious that this is a sphere of radius k, and for b) i see that the cosine of the angle of r vector and u hat is a constant so i think this will lead to a cone.
for c) i am not sure since i get the cosine of the angle between r and uhat is ranging between -1 and 1 so the angle is between 0 and pi, but considering the variability in magnitude of r i am not sure what this defines, and for d) i am not sure what to do

any help would be appreciated
 
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  • #2
no one can help with this?
 
  • #3
thenewbosco said:
no one can help with this?

In d), think about what the vector [tex](\vec{r}\cdot\hat{u})\hat{u}[/tex] represents.
 
  • #4
for d) this is just r minus the projection of r in the uhat direction? how should i interpret this? as a line?

and also i am not sure how to interpret what i have found in part c) if i could get some explanation it would be appreciated...thanks
 
  • #5
Regarding c) - you get that the cosine of the angle between r and u equals m. It's obvious that's some kind of cone. I only didn't understand the -1 to 1 part, but nevermind, you should be on the right track by now.
 
  • #6
i got that b) was a cone and it should be m goes from -1 to 1, so that means that the angle can vary from 0 to pi, so i am not sure how to look at this. Like r vector can be any direction from 0 to pi with the u unit vector, but given that r can vary in length it seems to be that it will be "half of all of space" from 0 to pi...
 
Last edited:
  • #7
if it is just the unit vector going from 0 to pi then this is a half sphere i guess, could this be it?
 
  • #8
Decide what is going from 0 to Pi, i.e. what does -1 <= 1 in c) mean?
 
  • #9
i am not sure what is going from 0 to pi, i thought initially it was the r vector, but i don't think its correct. if it is the unit vector this would make more sense
 

Related to Identifying Surfaces for Vectors: k, l, m, n, \hat{u}

1. What are vectors k, l, m, n, and ^u used for in identifying surfaces?

Vectors k, l, m, n, and ^u are used to represent the direction and magnitude of different forces acting on a surface. They are often used in physics and engineering to analyze and calculate the behavior of objects on surfaces.

2. How do you determine the direction of vector k, l, m, n, or ^u?

The direction of a vector is determined by the angle at which it is pointing. This can be calculated using trigonometric functions or by using the direction cosines of the vector, which represent the ratios of its components in different axes.

3. What is the difference between a vector and a scalar?

A vector has both magnitude and direction, while a scalar only has magnitude. In the context of identifying surfaces, vectors k, l, m, n, and ^u are used to represent forces with both magnitude and direction, while scalars may represent things like temperature or pressure, which do not have a specific direction.

4. How do you use vectors k, l, m, n, and ^u to analyze a surface?

Vectors k, l, m, n, and ^u can be used in various calculations and equations to analyze the forces acting on a surface. They can also be used to determine the normal vector of a surface, which is important in understanding its orientation and behavior.

5. Can vectors k, l, m, n, and ^u be used in any coordinate system?

Yes, vectors can be represented in any coordinate system, such as Cartesian, polar, or cylindrical coordinates. However, the calculations and equations used with these vectors may differ depending on the coordinate system being used.

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