New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

Be part of our community, it's free to join!

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

MATH 241 Week 1 Notes (12.1-12.2)

by: Tori Colthurst

MATH 241 Week 1 Notes (12.1-12.2) MATH 241

Tori Colthurst
View Full Document for 0 Karma

View Full Document


Unlock These Notes for FREE

Enter your email below and we will instantly email you these Notes for Calculus III

(Limited time offer)

Unlock Notes

Already have a StudySoup account? Login here

Unlock FREE Class Notes

Enter your email below to receive Calculus III notes

Everyone needs better class notes. Enter your email and we will send you notes for this class for free.

Unlock FREE notes

About this Document

These notes cover section 12.1 and 12.2 and were taken in Professor Whittlesey's Lecture.
Calculus III
Kim Whittlesey
Class Notes
Math, Calculus, 3, III, Vector, function, Plane, planes, vectors, scalar, multiplication, Dot, Product, unit, projection, normal, equation, line




Popular in Calculus III

Popular in Mathematics

This 6 page Class Notes was uploaded by Tori Colthurst on Thursday August 11, 2016. The Class Notes belongs to MATH 241 at University of Illinois at Urbana-Champaign taught by Kim Whittlesey in Fall 2016. Since its upload, it has received 49 views. For similar materials see Calculus III in Mathematics at University of Illinois at Urbana-Champaign.


Reviews for MATH 241 Week 1 Notes (12.1-12.2)


Report this Material


What is Karma?


Karma is the currency of StudySoup.

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 08/11/16
Calculus III (Professor Whittlesey) :​ Week 1 Notes (12.1­12.2)    ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­  Review  ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­        f’(a) = slope of tangent line @ x=a          b    ∫f(x) dx = area under the curve  a         b Fundamental Theorem of Calculus:​    f(x) dx = F(b)∫­ F(a)  a when F’(x)=f(x)  Function:    Function analogy : function=machine  Input 2 , output 4        More Variables: Input mult. variables, output one answer  p(r,c,t,etc) is a multivariable function      Input time, output parameterized equations              ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­  3​ n​ 12.1 Geometry of R​ ,....R​   ​ (R representing the notation for dimension)  ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­  R​  = 2 dimensional space (x,y) R​  = 3 dimensional space   (x,y,z)              Distance:    Distance from (0,0) to (x,y) → d=  x +y    2 2 √ Distance from (0,0,0) to (x,y,z) → d=  x +y +z    2 2 2 √ Distance from (a,b,c) to (x,y,z) →  ​**d=  (x−a) +(y−b) +(z−c)  **  2 √ R​  ? → add another term (ie. (w­d)​ ) for each additional dimension    2​ 2​ x​ +y​ =4 when graphed:  Circle radius 2 set of all points in R​  that satisfy x​ +y​ =4 2​ 2​ Center (0,0)     OR  set of all points in R​  that have distance 2   from the origin (0,0)    Sphere distance from (x,y,z) to (0,0,0)  Radius r √ x +y +z   = radius   2​ 2​ 2​ 2​ 3  Center (0,0,0) x​ +y​ +z​ =r​   eq. for a sphere in R​   distance from (x,y,z) to (a,b,c)   (x−a) +(y−b) +(z−c)   = 2 ​  radius  √ **(​ ­a)​ +(y­b)​ +(z­c)​ =r​ ** e​ q. for sphere in R​   3 Lines/Planes:     x/2 + y/3 = 1 imagine a plane that slices   through points  x/2+y/3+z/4=1  6x+4y+3z=12  General form for a plane in R​   **Ax+By+Cz=D**  All terms linear (no roots, squares, etc) therefore the “object is flat”     ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­ ­­­­­­­­­ 12.2 Vectors  (Scalar mult. / addition / unit vectors / dot product)   ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­  *vectors will be denoted by a ‘ (ex. v’)*    Vector:  ​    has 2 parts   1) magnitude (#)   2) direction  (imagine it as an arrow shaped sticky note that you can   pick up and put anywhere on the graph)  If v’ and w’ are the same length and direction except for  transition then v’=w’. (v/w being vectors)    algebraically           v’ = <a,b>  (always over a, up b wherever it is on plane)       Length of v = |v’| =  a√+b    2       Scalar multiplication   c*v’ =  c<a,b>  = <ca,cb> c→ scalar(#)  v’→vector : referred to as “scalar times vector”  The vector points in t ​ he same direction  ​ if c>0  the opposite direction​ if c<0  If c=0 the vector will be “squished” (no longer has magnitude)    Vector addition  Graphically:  *attach tail to       Head      Algebraically:  u’=<a,b>  v’=<c,d>  u’+v’=<a+c,b+d>  (Add x coordinates and y coordinates)  Commutative  u’+v’ = v’+u’  Associative (u’+v’)+w’=u’+(v’+w’)  Distributive c(u’+v’)=cu’+cv’    Unit vector:​ is a vector of length 1   In R​  (3 dimensions) vectors have 3 coordinates <a,b,c>    3 Special “basis” vectors for R​     i = <1,0,0> unit vector in each direction  j = <0,1,0>   ex)  <2,3,4> = 2<1,0,0>+3<0,1,0>+4<0,0,1> = 2i+3j+4k  k = <0,0,1>      Puzzle :)   find a unit vector in same direction as <3,7,8>  2 2 2 Solution → 1) find vector length :   3 +7 +√   =  122   (too long for a unit vector!)  √         2) shrink it by scalar multiplication using 1/ 122    New vector:  1/ 122 * <3,7,8> = <3/ 122, 7/ 122, 8/ 122>  √ The new length will equal 1  In general: ​if length of |v| does not equal 0 then v’/|v| (vector divided by its length)  is a unit vector in the same direction    Two ways to multiply vectors (we will only learn dot product today)  1) Dot product (notation **)  :   v’ ** w’ (any R​ ) n​ 2) cross product v’ x w’ (only R​ )  3​   v’ = <a,b> dot product = v’**w’ = <a,b>**<c,d> = ac + bd    w’ = <c,d>  *****notice! The answer is a #!! Scalar!!  ex) <1,2> ** <3,5> = 3(1) + 5(2) = 13  The dot product of two vectors is a number/scalar    2 2​ 2​ 2​ The dot product gives you ​length  v’**v’ = |v’|​   (<a,b>**<a,b>=a​ +b​ =|v|​ )  And helps find ​angle between vectors v’**w’ = |v’||w’|cosθ   Derived from:     Law of cosines  2​ 2​ 2​  |w’­v’|​  = |v’|​  |w’|​  ­ 2|v’||w’|cosθ       ex) are <9,2> and <­2,9> perpendicular?  <9,2>**<­2,9> = 0   |v’||w’|cosθ =​ 0   cosθ= ​ 0   (ie they are perpendicular because cos(90) = 0)            ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­   12.2 Vectors  (Day 2)   ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­  Facts Summary:   v’**v’ = |v’|​   v’**w’ = |v’||w’|cos θ   Unit vector in same direction as v’ = v’/|v’|  If v’ and w’ are perpendicular then v’**w’ = 0    Vector projection    Break v’ into components  v’ = ai’ + bj’  ai’ is parallel to x­axis  bj’ is parallel to y­axis      Problem:   given vectors v’, w’ write w’ = u’ + t’, where u’ is parallel to v’ and t’ is perpendicular to v’    u’ = projection of w’ onto v’  →  ​proj​ w’​ (nov’​ion for projection)    Direction: same as v’   Length: |w’|cosθ    ​aka c ​ omp​ wv’​ ​ (notation for length)  Formula: proj​ w’ v’​|w’|cosθ ) ​ *(v’/|v’|)  (length* unit vector)  Turns into (v’**w’ / v’**v’)(v’) = vector             number(vector)    ex) w’ = <1,3>         v’ = <5,3>      proj​v’​ = (v’**w’ / v’**v’) (v’) = 14/34<5,3> = <13/17, 21/17>      L’ lift, perpendicular to wing  W’ weight  → need v’ and w’ to balance  So project L’ onto W’ to get v’ v’ = proj​ L’  w’​   Planes  A line is uniquely determined by 2 points  A plane is uniquely determined by 3 points (not all on the same line)  Better characterization → use normal vector and a point     p = (a,b,c) point on plane    N’ = <d,e,f> normal vector (perpendicular to plane)(any location okay)        Suppose Q = (x,y,z) is another point on the plane  Vector from p to Q → <x­a,y­b,z­c>   The dot product between the vector formed by the points and the normal vector  will be 0 because they are perpendicular to one another  <x­a,y­b,z­c> ** <d,e,f> = 0  d(x­a) + e(y­b) + f(z­c) = 0  equation for a plane    ex) 3(x­1) + 2y ­ 5(z+7) = 0        N’ <3,2,­5>         P (1,0,­7)        Q (0,0,­38/5)   (another point by plugging in (0,0,x))      Find the angle between the planes (b/t 0 and pi/2)    S: 2x­z=5 T: x+5y­z=11     N’S​ = <2,0,­1>   N’​T​= <1,5,­1>    Find the angle between the normal vectors   2(1) + 0(5) + ­1(­1) = ( 5)( 27) cosθ   Solve for θ  arccos(3/ 135)  Lines  equation for a line:   ­­ intersection of 2 planes  ­­ parameterized line  Parameterized line:  ­­ pick a point p=(a,b,c) and a direction vector v’=<d,e,f>  (x,y,z) = (a,b,c) + t(d,e,f) for all t between (­infinity, infinity)    3​ In R​  lines can be:  parallel  (direction v’)           meet (point)           skew (not parallel or meeting) 


Buy Material

Are you sure you want to buy this material for

0 Karma

Buy Material

BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.


You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Kyle Maynard Purdue

"When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the I made $280 on my first study guide!"

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Parker Thompson 500 Startups

"It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!

Refund Policy


All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email


StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here:

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.