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Do there exist scalars k and l such that the vectors u = (2,k,6) , v = (l,5,3) , and w = (1,2,3) are mutually orthogonal with respect to the Euclidean inner product?
 one year ago
 one year ago
Do there exist scalars k and l such that the vectors u = (2,k,6) , v = (l,5,3) , and w = (1,2,3) are mutually orthogonal with respect to the Euclidean inner product?
 one year ago
 one year ago

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KonradZuseBest ResponseYou've already chosen the best response.0
The answer says no, but I originally thought that since k and l were u2 and v1 that itonly matters the 3rd spot, but I realized that you do (u1v1w1) +etc....
 one year ago

KonradZuseBest ResponseYou've already chosen the best response.0
Isn't it possible that we could find something = 0? Since that is what orthagonal means.
 one year ago

KonradZuseBest ResponseYou've already chosen the best response.0
Oh weait can scalars be negative?
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
if all three vectors were orthogonal, then:\[u\cdot w = 0\Longrightarrow (2)(1)+(k)(2)+(6)(3)=0\]\[2+2k+18=0\Longrightarrow 2k=20\Longrightarrow k=10\]Similarly,since v and w are orthogonal, we get that l must be 19.
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
yes scalars can be negative.
 one year ago

KonradZuseBest ResponseYou've already chosen the best response.0
So we are actually solving for them...? I thought it was any number..
 one year ago

KonradZuseBest ResponseYou've already chosen the best response.0
or those would be the numbers to = 0?
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
Since we are asking "does there exist", the question is asking "is there any one such number k and l." Its not the same as "for all/any k and l."
 one year ago

KonradZuseBest ResponseYou've already chosen the best response.0
It doens't say 1 though? Or does that "mutually orthagonal" mean something?
 one year ago

KonradZuseBest ResponseYou've already chosen the best response.0
Do there exist scalars k and l such that the vectors
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
If u and w are orthogonal, then k would have to be 10. If k is any other number, they wont be orthogonal since the inner/dot product wouldnt come out to zero.
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
mutually orthogonal means that all three vectors are perpendicular to each other.
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
The thing is, there is no way these three vectors can be mutually orthogonal. Since k would have to be 10, and l would have to be 19. Then u and v wouldnt be orhtogonal. There is no way to get all three to be perpendicular at the same time.
 one year ago

KonradZuseBest ResponseYou've already chosen the best response.0
I see, makes sense, thanks!
 one year ago
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