1. Not finding help here? Sign up for a free 30min 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!

Linear Algebra- Vector proof

  1. Jan 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that if a vector u, is perpendicular to both v and w, then u is also perpendicular to v+w. More generally, show that u (perp) (sv+tw) for all scalars s and t.


    2. Relevant equations
    I was thinking that the cross product would be relevant.


    3. The attempt at a solution
    I'm not quite sure where to start. I tried to use w[-wy, wx] and so forth, but I honestly don't understand their full meanings. Any help would be appreciated
     
  2. jcsd
  3. Jan 11, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I think you ought to be thinking about the dot product. How do you express the notion two vectors are perpendicular in terms of the dot product?
     
  4. Jan 11, 2009 #3

    rock.freak667

    User Avatar
    Homework Helper

    If u is perpendicular to v and w, then v and w should lie on the same plane, correct? Thus if v+w, is also perpendicular to u, then what can you say about the vectors, v,u and v+w ?


    EDIT: I believe Dick's method is more feasible than what I was trying to do.
     
  5. Jan 11, 2009 #4
    that u.(v+w)=0?
     
  6. Jan 11, 2009 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Right. u.(v+w)=0 says u and v+w are perpendicular. Does that follow from u.v=0 and u.w=0?
     
  7. Jan 11, 2009 #6
    Yes? Because both v and w would have to be perpendicular to u? I'm not really sure though...
     
  8. Jan 11, 2009 #7

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    The problem says "u is perpendicular to both v and w". That means u.v=0 and u.w=0. What can you conclude about u.(v+w) and why?
     
  9. Jan 11, 2009 #8
    I really don't know.. I'm sorry. I don't think that I have enough background knowledge, we've only had one lecture.
     
  10. Jan 11, 2009 #9

    rock.freak667

    User Avatar
    Homework Helper

    Check the dot product's properties
     
  11. Jan 11, 2009 #10
    Write out u.v + u.w and u.(v+w)
     
  12. Jan 11, 2009 #11
    u.v=-u.w? Because of distributive laws?
     
  13. Jan 11, 2009 #12

    rock.freak667

    User Avatar
    Homework Helper

    By the distributive laws, can you expand u.(v+w)?
     
  14. Jan 11, 2009 #13
    You should be able to see that u.v + u.w = u.(v+w) just by writing out the terms on each side
     
  15. Jan 11, 2009 #14

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    No. Because of distributive law u.(v+w)=u.v+u.w. Doesn't that look more like a 'distributive law'?
     
  16. Jan 11, 2009 #15
    right...
     
  17. Jan 11, 2009 #16
    but we know that u.(v+w)=0 . Wouldn't that imply that u.v+u.w=0?
     
  18. Jan 11, 2009 #17

    rock.freak667

    User Avatar
    Homework Helper

    Actually, you knew that u.(v+w)=u.v + u.w, from the question, you deduced that u.v=0 and u.w=0
    So u.(v+w)=0, which means?
     
  19. Jan 11, 2009 #18
    that the vector u is perpendicular to v+w?
     
  20. Jan 11, 2009 #19

    rock.freak667

    User Avatar
    Homework Helper

    Yes.

    Now you want to prove that u is perpendicular to (sv+tw) for all scalars s and t.

    Now consider what u.(sv+tw) expands out to be.
     
  21. Jan 11, 2009 #20

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You don't know u.(v+w)=0. That's what you are trying to prove. What you know is that u.v=0 and u.w=0.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Linear Algebra- Vector proof
  1. Linear algebra proofs (Replies: 14)

  2. Linear Algebra (Replies: 2)

Loading...