## Review for Final Exam—Part 1

- Find the angle between the vectors and
- Find the curvature at of the curve
- Find parametric equations for the tangent line at the point to the curve with parametric equations
- Find the length of the curve for
- Find an equation of the plane that contains the point and is perpendicular to the line with equation
- Find an equation of the plane that contains the line parallel to the plane
- Find the distance from the point to each of the following:
- The plane
- The axis.

- Find the volume of a parallelepiped with adjacent edges where
- At what points does the helix intersects the sphere
- Find a non-zero vector orthogonal to the plane through the points

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.

Go for it! I think it is a brilliant idea

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??

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.

thanks, I was wondering about this one also!

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?

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

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

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 🙂

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.

perpendicular* (not parallel)

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

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.

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

hint.. look for factor the polynomial

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.

Refer to above reply.

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

Refer to above reply about this same question.

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

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.

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.

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

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

Okay thanks!

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

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

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.

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.

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

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.

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.

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.

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

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

can we use formula sheet on final?

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

He didn’t change his mind.

For the curvature equation what exactly is T?

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

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

My bad! I took care of the typo already.

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

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

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!