1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Orthogonal and Parallel Vectors

  1. Oct 4, 2014 #1
    1. The problem statement, all variables and given/known data
    For the following vectors 'u' and 'v', express 'u' as the sum u = p + n where 'p' is parallel to 'v' and 'n' is orthogonal to 'v'
    u = {-1, 2, 3}
    v = {2, 1, 1}

    2. Relevant equations
    Dot product
    Cross product

    3. The attempt at a solution
    First, I should say that I do not know how to use latex for determinants, so if somebody can help me out there I could make this problem a bit easier to read. Now, I am aware that two vectors are orthogonal if their dot product is 0 and parallel if their cross product is zero. I used this knowledge to attempt to solve the problem:
    0 = p x v
    0 = n 'dot' v
    To attempt to figure out the value of p={a, b, c} I performed the cross product and obtained:
    [itex] i (c-b) - j (2c-a) + k (2b-a) = 0 [/itex]
    [itex] c = b [/itex]
    [itex] 2c = a [/itex]
    [itex] 2b = a [/itex]
    Next, to attempt to figure out the value of n={x, y, z} I performed a the dot product and obtained:
    [itex] (2, 1, 1) dot (x, y, z) [/itex]
    [itex] 2x + y + z = 0 [/itex]
    At this point I am not sure what to do, if somebody could give me some advice I'd greatly appreciate it.
     
  2. jcsd
  3. Oct 4, 2014 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You are thinking too hard. cv is parallel to v for any choice of scalar c. Let that be parallel part. Then the other part of the sum must be u-cv. Find c by making sure u-cv is perpendicular to v using the dot product.
     
  4. Oct 4, 2014 #3
    Ok so I made c =2:
    cv = {4, 2, 2} = p
    n = u - p = {-1, 2, 3} - {4, 2, 2} = {-5, 0 , 2}
    u = p + n = {4, 2, 2} + {-5, 0 ,2} = {-1, 2, 4}
    is the problem really this simple? XD
     
  5. Oct 4, 2014 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, it's basically that simple. But I don't get c=2. Your n isn't perpendicular to v. How did you get that?
     
  6. Oct 4, 2014 #5
    It was arbitrary, I thought two vectors are parallel so long as their components are proportional, {4, 2, 2} is proportional to {2, 1, 1}, no?
     
  7. Oct 4, 2014 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, but u-cv needs to be perpendicular to v as well. That's what determines c.
     
  8. Oct 4, 2014 #7
    ohhh...ok, let me try again..
     
  9. Oct 4, 2014 #8
    nvm I thought I knew how to do it but I dont....would I find c by:
    {-1, 2, 3} - c{2, 1, 1} = 0 ?
     
  10. Oct 4, 2014 #9

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    No. u-cv is supposed to perpendicular to v. Use the dot product! So (u-cv).v=0. Solve that.
     
  11. Oct 4, 2014 #10
    sorry I thought I solved it but made a mistake....trying again
     
  12. Oct 4, 2014 #11
    Ok I got c=1/2, is that correct?
     
  13. Oct 4, 2014 #12
    sorry, I should show how I got that....
    (u-cv).v = 0
    u.v - cv.v = 0
    {-1,2,3}.{2,1,1} - c{2,1,1}.{2,1,1} = 0
    3 - 6c = 0
    c = 1/2
     
  14. Oct 4, 2014 #13

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, c=1/2 not 2.
     
  15. Oct 4, 2014 #14

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Just to state, about the general case, that in any vector space V where orthogonality makes sense and W is a subspace of V , we have ## V= W \oplus W^{\perp} ## then every vector can be written as the sum of a vector w in W and a vector ## w^{\perp}## in ## W^{\perp} ##. This also follows from the Fundamental Theorem of Linear Algebra.
     
  16. Oct 4, 2014 #15
    thank you for your help...youre right, i was thinking to much about that one XD
     
  17. Oct 4, 2014 #16
    sd
    I appreciate your help but I'm not familiar with the FToLA (perhaps you could elaborate)....I plan on taking Linear Algebra next semester :D
     
  18. Oct 4, 2014 #17

    WWGD

    User Avatar
    Science Advisor
    Gold Member

  19. Oct 4, 2014 #18
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Orthogonal and Parallel Vectors
  1. Parallel Vectors (Replies: 4)

  2. Orthogonal vector (Replies: 7)

  3. Parallel vectors (Replies: 4)

  4. Parallel Vectors (Replies: 13)

Loading...