Vectors and surfaces

1. Nov 8, 2006

thenewbosco

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

2. Nov 11, 2006

thenewbosco

no one can help with this?

3. Nov 11, 2006

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

4. Nov 11, 2006

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

5. Nov 11, 2006

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. Nov 11, 2006

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...

Last edited: Nov 11, 2006
7. Nov 11, 2006

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?

8. Nov 11, 2006