## Review for Final Exam—Part 1

1. Find the angle between the vectors $v=\langle 2,0,1 \rangle$ and $w=\langle 4,-2,0 \rangle.$
2. Find the curvature at $t=0$ of the curve $\boldsymbol{r}(t) = \langle \cos(2t), 4t, \sin(2t) \rangle.$
3. Find parametric equations for the tangent line at the point $(5,0,2)$ to the curve with parametric equations
$\begin{cases} x(t) =3+2\sqrt{t} \\ y(t)=t^3-t\\ z(t) = t^3+t \end{cases}$
4. Find the length of the curve $\boldsymbol{r}(t) = 6t \boldsymbol{i} + 8t^{3/2} \boldsymbol{j} + 6t^2 \boldsymbol{k},$ for $0 \leq t \leq 1.$
5. Find an equation of the plane that contains the point $(1,3,5)$ and is perpendicular to the line with equation $\big\{ x=3+t, y=3t, z=5-2t \big\}.$
6. Find an equation of the plane that contains the line $\big\{ x=3+4t, y=2, z=4t \big\}$ parallel to the plane $3x+6y-3z=18.$
7. Find the distance from the point $(3,7,-5)$ to each of the following:
• The $xy-$plane
• The $y-$axis.
8. Find the volume of a parallelepiped with adjacent edges $PQ, PR, PS,$ where $P=(3,0,3), Q=(-1,3,6), R=(5,3,1), S=(2,4,4).$
9. At what points does the helix $\boldsymbol{r}(t)= \langle \sin(t), \cos(t), t \rangle$ intersects the sphere $x^2+y^2+z^2=10?$
10. Find a non-zero vector orthogonal to the plane through the points $P=(-2,0,0), Q=(0,3,0), R=(0,0,1).$
1. December 2, 2012 at 10:52 pm

even though this is my groups part.. i believe some of these questions could have had a multiple part question that requires several basic concepts to solve the more complex concept.

• December 2, 2012 at 10:53 pm

Go for it! I think it is a brilliant idea

2. December 3, 2012 at 3:53 pm

For #3 we just find the r'(t) and plug in our t value to get the tangent and then just change it to parametric equations correct??

• December 3, 2012 at 10:21 pm

First solve the original parametric equations for t. My suggestion is to solve x(t). Check to make sure that this ‘t’ satisfies the other parametric equations. Next take the derivative of r(t) and plug in the found t value. This is your (a,b,c). Remember that the formula for parametric equations are x=x0 + t x a, etc. Let me know if you have any more questions.

• December 5, 2012 at 12:21 pm

3. December 3, 2012 at 8:12 pm

For number six, don’t I make two vectors and do the dot product and if it equals zero it is parallel correct? Seeing that is true, I can then make the equation correct?

• December 3, 2012 at 10:24 pm

Yes, it seems like you are on the right path. If you continue to have trouble let me know.

• December 10, 2012 at 9:28 am

what points do we use to make the equation? I used (3,2,0) and got the final equation to be 3(x-3)+ 6(y-2)-3(z)=0. is that correct

• December 4, 2012 at 1:06 am

Sorry you are right for dot product if it equals 1 it is parallel but for number 6 the dot product for this vector is 0 so therefore, solve it so it is perpendicular to the other plane. My fault guys 🙂

• December 10, 2012 at 2:59 pm

Actually you do want the two planes to be parallel. The normal vector of the plane that is given perpendicular to the plane itself. Since this vector is perpendicular to the plane, it would also be perpendicular to the given line. If these two are perpendicular then you can use the dot product set equal to zero, because when the dot product is zero the two vectors given are perpendicular.

4. December 3, 2012 at 8:15 pm

perpendicular* (not parallel)

5. December 3, 2012 at 8:25 pm

And for number four, were suppose to take the integral from 0 to 1 of the derivative of the equation correct? I did that and got the integral from 0 to 1 of sqrt(36+144t+144t^2) is that correct? If so, how am i suppose to take the integral of that

• December 3, 2012 at 10:27 pm

Yes, you do take the derivative. After you take the derivative though you must find the magnitude. Once you have the magnitude of the dervative then integrate over that between 0 and 1.

• December 3, 2012 at 11:49 pm

Gaston you have to simplify to get rid of the square root.. once you have it with out the square root it will be so much easier

• December 3, 2012 at 11:52 pm

hint.. look for factor the polynomial

6. December 3, 2012 at 9:38 pm

Yea Im having the same problem with #4 as Gaston here…the integral of 6sqrt((1+2t)^(2)). I typed it into wolfram but i can’t figure out how to solve it to get what they have. So if someone could just provide a little hint it would help. like what to use cause I tried u sub but im either doing it wrong or thats not it.

• December 3, 2012 at 10:28 pm

7. December 3, 2012 at 9:52 pm

For number 3 question , I formed a vector between the point (5,0,2) and (0,0,0). What should I do next?

• December 3, 2012 at 10:28 pm

8. December 3, 2012 at 10:40 pm

Keep in mind that for Number 2 there are two equations for curvature that you can use to solve the problem.

9. December 3, 2012 at 10:46 pm

For Number 4, You take the derivative of the equation, then find the magnitude of the derivative of r. after that, you integrate from 0-1.

10. December 3, 2012 at 11:42 pm

Abibatu, for number 3 plug in the points from the line for x, y, and z for the parametric equation of the curve. then solve for t. next take the derivative of the vector from the given parametric equations. then plug t back in to find another set of points. now you can set up the parametric equations using the given points and the points you just found.

11. December 3, 2012 at 11:45 pm

for number 4… take the derivative of r(t) then the magnitude.. then preform an integral using the given bounds.

12. December 3, 2012 at 11:47 pm

Gaston! you got it but now simplify to get rid of the square root…once you simplified it to without the square root it will be so much easier

• December 4, 2012 at 7:45 pm

Okay thanks!

13. December 5, 2012 at 10:46 am

Can someone remind me how to start number 9? I think I know how, but I can’t seem to find it in my notes to be sure

14. December 5, 2012 at 11:04 am

For number 9, do I just plug in the values into the sphere?

• December 5, 2012 at 11:11 am

plug x(t), y(t), and z(t) into the sphere and solve for t, not that they must be the say t. then plug that t value(s) into r(t) to find the points.

15. December 5, 2012 at 11:33 am

What was the final answer for number 2? I did lr'(t) x r”(t)l / lr'(t)^3l and got one but i’m not sure if it is correct.

• December 12, 2012 at 6:50 pm

i did the other equation and got 37^3/2

16. December 5, 2012 at 8:25 pm

Is there any way somebody could confer answers with me for number 2? I have done it 4 different times now, but I got a different answer each time.

17. December 5, 2012 at 8:28 pm

Also, or number four, aren’t we supposed to take the derivative of the equation, then take the derivative of that from 0-1? I did that, and I got a really difficult integral. I computed it with my calculator but it gave a strange result.

• December 7, 2012 at 11:49 am

For number 4 you should first take the derivative. Once getting this answer take the magnitude. The magnitude with come out to be a function squared under the square root sign. Then take the integral of the magnitude with respect to t and between the values 0 and 1.

18. December 6, 2012 at 8:29 pm

For number ten do we only have to do the cross product of the directional vector and calculate the magnitude?

• December 7, 2012 at 11:53 am

First find two vectors from those points. Then calculate the cross product.

19. December 6, 2012 at 11:32 pm

can we use formula sheet on final?

• December 12, 2012 at 5:39 pm

Dr.Blanco-Silva said that we could in class so unless he changed his mind we still can

• December 12, 2012 at 8:46 pm

He didn’t change his mind.

20. December 11, 2012 at 3:21 am

For the curvature equation what exactly is T?

• December 12, 2012 at 11:07 am

The vector “T(t)” is the derivative of the vector “r” over the magnitude of the derivative of vector “r.”

21. December 12, 2012 at 11:22 am

is number 9 supposed to be x^2+y^2+x^2=10 or is that last x supposed to be a z??

• December 12, 2012 at 11:24 am

• December 12, 2012 at 11:28 am

Thank you! that will make it come out a little better

22. December 12, 2012 at 6:40 pm

Does anyone have suggestions to help with visualization of finding lines through certain points, planes, etc.

23. December 12, 2012 at 6:49 pm

For question number 8, I just wanted to clarify that the correct equation to use is the magnitude of a dot b x c? sorry if that’s a little confusing to read, but I just wanted to be sure!