Magnitude of the sum of vectors in 3D space

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I have three vectors $u_1,u_2,u_3$,



$$
||u_1||=3\
||u_2||=1\
||u_3||=2\
angle u_1,u_2=fracpi3\
angle u_3,u_1=fracpi4\
angle u_3,u_2=fracpi4.
$$



I am asked to calculate the length of $u_1+u_2+u_3$ and the area of the parallellogram spanned by $u_1+u_2$ and $u_3$.



My problem is with calculating the length of the vector sums. I don't believe I can use the normal pythagorean theorem since the vectors aren't all orthogonal to each other, but I assume I could use some generalized version of the Pythagorean theorem. However, since we haven't really learned that in the course I am taking, I don't think that's the point. Is there another way to do this using the most elementary concepts of linear algebra such as dot products, vector products etc.? I have a feeling the answer is really obvious but I just don't see it.




linear-algebra vectors




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    up vote
    0
    down vote

    favorite












    I have three vectors $u_1,u_2,u_3$,



    $$
    ||u_1||=3\
    ||u_2||=1\
    ||u_3||=2\
    angle u_1,u_2=fracpi3\
    angle u_3,u_1=fracpi4\
    angle u_3,u_2=fracpi4.
    $$



    I am asked to calculate the length of $u_1+u_2+u_3$ and the area of the parallellogram spanned by $u_1+u_2$ and $u_3$.



    My problem is with calculating the length of the vector sums. I don't believe I can use the normal pythagorean theorem since the vectors aren't all orthogonal to each other, but I assume I could use some generalized version of the Pythagorean theorem. However, since we haven't really learned that in the course I am taking, I don't think that's the point. Is there another way to do this using the most elementary concepts of linear algebra such as dot products, vector products etc.? I have a feeling the answer is really obvious but I just don't see it.




    linear-algebra vectors




    share|cite|improve this question






















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I have three vectors $u_1,u_2,u_3$,



      $$
      ||u_1||=3\
      ||u_2||=1\
      ||u_3||=2\
      angle u_1,u_2=fracpi3\
      angle u_3,u_1=fracpi4\
      angle u_3,u_2=fracpi4.
      $$



      I am asked to calculate the length of $u_1+u_2+u_3$ and the area of the parallellogram spanned by $u_1+u_2$ and $u_3$.



      My problem is with calculating the length of the vector sums. I don't believe I can use the normal pythagorean theorem since the vectors aren't all orthogonal to each other, but I assume I could use some generalized version of the Pythagorean theorem. However, since we haven't really learned that in the course I am taking, I don't think that's the point. Is there another way to do this using the most elementary concepts of linear algebra such as dot products, vector products etc.? I have a feeling the answer is really obvious but I just don't see it.




      linear-algebra vectors




      share|cite|improve this question












      I have three vectors $u_1,u_2,u_3$,



      $$
      ||u_1||=3\
      ||u_2||=1\
      ||u_3||=2\
      angle u_1,u_2=fracpi3\
      angle u_3,u_1=fracpi4\
      angle u_3,u_2=fracpi4.
      $$



      I am asked to calculate the length of $u_1+u_2+u_3$ and the area of the parallellogram spanned by $u_1+u_2$ and $u_3$.



      My problem is with calculating the length of the vector sums. I don't believe I can use the normal pythagorean theorem since the vectors aren't all orthogonal to each other, but I assume I could use some generalized version of the Pythagorean theorem. However, since we haven't really learned that in the course I am taking, I don't think that's the point. Is there another way to do this using the most elementary concepts of linear algebra such as dot products, vector products etc.? I have a feeling the answer is really obvious but I just don't see it.





      linear-algebra vectors





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      share|cite|improve this question




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      asked Aug 16 at 10:20









      Chisq

      1467




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          By $ cdot $ I denote the usual scalar product in $3D$.



          Hints:



          1. We have $u_i cdot u_j =||u_i||||u_j|| cos angle u_i,u_j$.


          2. $||u_1+u_2+u_3||^2=(u_1+u_2+u_3) cdot (u_1+u_2+u_3) $.


          Can you proceed ?






          share|cite|improve this answer




















          • Ah, of course! I guess it was just as obvious as I imagined it would be. Thanks a lot.
            – Chisq
            Aug 16 at 10:28










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          1 Answer
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          active

          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          2
          down vote



          accepted










          By $ cdot $ I denote the usual scalar product in $3D$.



          Hints:



          1. We have $u_i cdot u_j =||u_i||||u_j|| cos angle u_i,u_j$.


          2. $||u_1+u_2+u_3||^2=(u_1+u_2+u_3) cdot (u_1+u_2+u_3) $.


          Can you proceed ?






          share|cite|improve this answer




















          • Ah, of course! I guess it was just as obvious as I imagined it would be. Thanks a lot.
            – Chisq
            Aug 16 at 10:28














          up vote
          2
          down vote



          accepted










          By $ cdot $ I denote the usual scalar product in $3D$.



          Hints:



          1. We have $u_i cdot u_j =||u_i||||u_j|| cos angle u_i,u_j$.


          2. $||u_1+u_2+u_3||^2=(u_1+u_2+u_3) cdot (u_1+u_2+u_3) $.


          Can you proceed ?






          share|cite|improve this answer




















          • Ah, of course! I guess it was just as obvious as I imagined it would be. Thanks a lot.
            – Chisq
            Aug 16 at 10:28












          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          By $ cdot $ I denote the usual scalar product in $3D$.



          Hints:



          1. We have $u_i cdot u_j =||u_i||||u_j|| cos angle u_i,u_j$.


          2. $||u_1+u_2+u_3||^2=(u_1+u_2+u_3) cdot (u_1+u_2+u_3) $.


          Can you proceed ?






          share|cite|improve this answer












          By $ cdot $ I denote the usual scalar product in $3D$.



          Hints:



          1. We have $u_i cdot u_j =||u_i||||u_j|| cos angle u_i,u_j$.


          2. $||u_1+u_2+u_3||^2=(u_1+u_2+u_3) cdot (u_1+u_2+u_3) $.


          Can you proceed ?







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 16 at 10:26









          Fred

          38k1238




          38k1238











          • Ah, of course! I guess it was just as obvious as I imagined it would be. Thanks a lot.
            – Chisq
            Aug 16 at 10:28
















          • Ah, of course! I guess it was just as obvious as I imagined it would be. Thanks a lot.
            – Chisq
            Aug 16 at 10:28















          Ah, of course! I guess it was just as obvious as I imagined it would be. Thanks a lot.
          – Chisq
          Aug 16 at 10:28




          Ah, of course! I guess it was just as obvious as I imagined it would be. Thanks a lot.
          – Chisq
          Aug 16 at 10:28












           

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