# 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