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

[thenewbosco]

The question reads:
Identify the following surfaces given that k, l, m, n are fixed values and \hat{u} is a fixed unit vector.

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

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

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

 

[thenewbosco]

no one can help with this?

 

[radou]

no one can help with this?

In d), think about what the vector (\vec{r}\cdot\hat{u})\hat{u} represents.

 

[thenewbosco]

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

 

[radou]

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.

 

[thenewbosco]

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...

 

[thenewbosco]

if it is just the unit vector going from 0 to pi then this is a half sphere i guess, could this be it?

 

[radou]

Decide what is going from 0 to Pi, i.e. what does -1 <= 1 in c) mean?

 

[thenewbosco]

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