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Introduction to linear algebra (Strang) 1.1 problem 20

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$begingroup$

Under what restrictions on c, d, e, will the combinations c$u$+d$v$+e$w$ fill the dashed triangle? ($u$, $v$, $w$ are 3-d vectors)

I have been trying to see the way I could make restrictions on c, d and e, but the only thing I can think of is getting vectors from the vectors' heads and using the norm of the cross product, but this is a general case and I think it won't work for every case (not sure).

Then it just doesn't makes sense for me because it is supposed to be a linear combination that fills spaces not a plane, then if it had the characteristics to fill the plane, shouldn't it be a plane lying on the vectors and not intercepted by them?

linear-algebra vectors

share|cite|improve this question

edited Mar 22 at 2:21

ricardovaras_99

asked Mar 22 at 2:11

ricardovaras_99ricardovaras_99

224

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you restrict to just $u$ and $v$, do you know for which scalars $a$ and $b$ that $au + bv$ lies in the line segment between $u$ and $v$?
  $endgroup$
  – Theo Bendit
  Mar 22 at 2:15
  

  
  
  
  

add a comment | 

0

$begingroup$

Under what restrictions on c, d, e, will the combinations c$u$+d$v$+e$w$ fill the dashed triangle? ($u$, $v$, $w$ are 3-d vectors)

I have been trying to see the way I could make restrictions on c, d and e, but the only thing I can think of is getting vectors from the vectors' heads and using the norm of the cross product, but this is a general case and I think it won't work for every case (not sure).

Then it just doesn't makes sense for me because it is supposed to be a linear combination that fills spaces not a plane, then if it had the characteristics to fill the plane, shouldn't it be a plane lying on the vectors and not intercepted by them?

linear-algebra vectors

share|cite|improve this question

edited Mar 22 at 2:21

ricardovaras_99

asked Mar 22 at 2:11

ricardovaras_99ricardovaras_99

224

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you restrict to just $u$ and $v$, do you know for which scalars $a$ and $b$ that $au + bv$ lies in the line segment between $u$ and $v$?
  $endgroup$
  – Theo Bendit
  Mar 22 at 2:15
  

  
  
  
  

add a comment | 

$begingroup$

Under what restrictions on c, d, e, will the combinations c$u$+d$v$+e$w$ fill the dashed triangle? ($u$, $v$, $w$ are 3-d vectors)

I have been trying to see the way I could make restrictions on c, d and e, but the only thing I can think of is getting vectors from the vectors' heads and using the norm of the cross product, but this is a general case and I think it won't work for every case (not sure).

Then it just doesn't makes sense for me because it is supposed to be a linear combination that fills spaces not a plane, then if it had the characteristics to fill the plane, shouldn't it be a plane lying on the vectors and not intercepted by them?

linear-algebra vectors

share|cite|improve this question

edited Mar 22 at 2:21

ricardovaras_99

asked Mar 22 at 2:11

ricardovaras_99ricardovaras_99

224

$endgroup$

share|cite|improve this question

edited Mar 22 at 2:21

ricardovaras_99

asked Mar 22 at 2:11

ricardovaras_99ricardovaras_99

224

share|cite|improve this question

edited Mar 22 at 2:21

ricardovaras_99

asked Mar 22 at 2:11

ricardovaras_99ricardovaras_99

224

asked Mar 22 at 2:11

ricardovaras_99ricardovaras_99

224

asked Mar 22 at 2:11

ricardovaras_99ricardovaras_99

224