Angle between vectors $vec a + vec b$ and $vec c$?

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If there are three vector $vec a$, $vec b$ and $vec c$ provided, what is the angle between $vec a + vec b$ and $vec c$? I understand how to calculate for angle between $vec a$ and $vec b$, $vec a$ and $vec c$ and $vec b$ and $vec c$ but what does the angle between $vec a + vec b$ and $vec c$ mean?




vectors inner-product-space




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    up vote
    2
    down vote

    favorite












    If there are three vector $vec a$, $vec b$ and $vec c$ provided, what is the angle between $vec a + vec b$ and $vec c$? I understand how to calculate for angle between $vec a$ and $vec b$, $vec a$ and $vec c$ and $vec b$ and $vec c$ but what does the angle between $vec a + vec b$ and $vec c$ mean?




    vectors inner-product-space




    share|cite|improve this question























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      If there are three vector $vec a$, $vec b$ and $vec c$ provided, what is the angle between $vec a + vec b$ and $vec c$? I understand how to calculate for angle between $vec a$ and $vec b$, $vec a$ and $vec c$ and $vec b$ and $vec c$ but what does the angle between $vec a + vec b$ and $vec c$ mean?




      vectors inner-product-space




      share|cite|improve this question













      If there are three vector $vec a$, $vec b$ and $vec c$ provided, what is the angle between $vec a + vec b$ and $vec c$? I understand how to calculate for angle between $vec a$ and $vec b$, $vec a$ and $vec c$ and $vec b$ and $vec c$ but what does the angle between $vec a + vec b$ and $vec c$ mean?





      vectors inner-product-space





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited 8 mins ago









      Jyrki Lahtonen

      104k12160355




      104k12160355









      asked 19 hours ago









      bison72

      152




      152




















          5 Answers
          5






          active

          oldest

          votes

















          up vote
          7
          down vote













          You compute the sum of $a+b$, that is a vector, if it helps, you can give a new name, $d$.



          Now you can compute the angle between $d$ and $c$ using the formula that you are familiar with.






          share|cite|improve this answer




























            up vote
            7
            down vote













            We have that $vec d=vec a+vec b$ then



            $$cos (theta)=fracvec d cdot vec c$$






            share|cite|improve this answer




























              up vote
              3
              down vote













              The important concept to keep in mind here is that when you add two vectors, $vec a$ and $vec b$, the result is just another vector. It has a magnitude, a direction (if its magnitude is positive), and components just like any other vector.



              We may write the expression
              $vec a + vec b$
              in order to name this vector, but that does not make it into any new kind of object.
              The formulas that apply to other vectors still apply.



              To find the angle between $vec a + vec b$ and $vec c,$
              take the formula you would use for the angle between any two vectors, put $vec a + vec b$
              in the places where the first vector occurs in the formula,
              and put $vec c$ in the places where the second vector occurs.
              Look at other answers for details of what happens after you do this.






              share|cite|improve this answer

















              • 1




                If vectors a=[1,1,0], b=[3,2,1], and c=[1,0,2], does the mean the angle between a +b and c will be the angle between [4,3,1] and [1,0,2]?
                – bison72
                14 hours ago






              • 1




                @bison72 yes, this is how vector addition works.
                – PaÅ­lo Ebermann
                9 hours ago

















              up vote
              2
              down vote













              Using the definition of the scalar product we get
              $$
              (a+b)cdot c =
              lVert a+b rVert lVert c rVert cos angle(a+b,c)
              $$
              where $lVert v rVert = sqrtvcdot v$.



              For non-zero $a+b$ and $c$ we can solve for the angle
              $$
              angle(a+b,c) = arccos frac(a+b)cdot clVert a+b rVert lVert c rVert
              $$






              share|cite|improve this answer






























                up vote
                2
                down vote













                Just $$cos(widehatveca+vecb,vecc)=frac(veca+vecb)vecc.$$



                Here, $|veca+vecb|=sqrt$ and $(veca+vecb)vecc=vecavecc+vecbvecc.$






                share|cite|improve this answer























                • Why someone down voted?
                  – Michael Rozenberg
                  14 hours ago










                • Might be because someone thought it's not obvious that e.g. $ab$ is the scalar product of vectors and not simple product of their norms (because you've dropped the arrows used by the OP). But that's just my guess.
                  – Ruslan
                  8 hours ago











                • @Ruslan But there is a context. Thank you for your trying! I'll fix it.
                  – Michael Rozenberg
                  1 hour ago











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                5 Answers
                5






                active

                oldest

                votes








                5 Answers
                5






                active

                oldest

                votes









                active

                oldest

                votes






                active

                oldest

                votes








                up vote
                7
                down vote













                You compute the sum of $a+b$, that is a vector, if it helps, you can give a new name, $d$.



                Now you can compute the angle between $d$ and $c$ using the formula that you are familiar with.






                share|cite|improve this answer

























                  up vote
                  7
                  down vote













                  You compute the sum of $a+b$, that is a vector, if it helps, you can give a new name, $d$.



                  Now you can compute the angle between $d$ and $c$ using the formula that you are familiar with.






                  share|cite|improve this answer























                    up vote
                    7
                    down vote










                    up vote
                    7
                    down vote









                    You compute the sum of $a+b$, that is a vector, if it helps, you can give a new name, $d$.



                    Now you can compute the angle between $d$ and $c$ using the formula that you are familiar with.






                    share|cite|improve this answer













                    You compute the sum of $a+b$, that is a vector, if it helps, you can give a new name, $d$.



                    Now you can compute the angle between $d$ and $c$ using the formula that you are familiar with.







                    share|cite|improve this answer













                    share|cite|improve this answer



                    share|cite|improve this answer











                    answered 19 hours ago









                    Siong Thye Goh

                    76.3k124592




                    76.3k124592




















                        up vote
                        7
                        down vote













                        We have that $vec d=vec a+vec b$ then



                        $$cos (theta)=fracvec d cdot vec c$$






                        share|cite|improve this answer

























                          up vote
                          7
                          down vote













                          We have that $vec d=vec a+vec b$ then



                          $$cos (theta)=fracvec d cdot vec c$$






                          share|cite|improve this answer























                            up vote
                            7
                            down vote










                            up vote
                            7
                            down vote









                            We have that $vec d=vec a+vec b$ then



                            $$cos (theta)=fracvec d cdot vec c$$






                            share|cite|improve this answer













                            We have that $vec d=vec a+vec b$ then



                            $$cos (theta)=fracvec d cdot vec c$$







                            share|cite|improve this answer













                            share|cite|improve this answer



                            share|cite|improve this answer











                            answered 19 hours ago









                            gimusi

                            63.4k73379




                            63.4k73379




















                                up vote
                                3
                                down vote













                                The important concept to keep in mind here is that when you add two vectors, $vec a$ and $vec b$, the result is just another vector. It has a magnitude, a direction (if its magnitude is positive), and components just like any other vector.



                                We may write the expression
                                $vec a + vec b$
                                in order to name this vector, but that does not make it into any new kind of object.
                                The formulas that apply to other vectors still apply.



                                To find the angle between $vec a + vec b$ and $vec c,$
                                take the formula you would use for the angle between any two vectors, put $vec a + vec b$
                                in the places where the first vector occurs in the formula,
                                and put $vec c$ in the places where the second vector occurs.
                                Look at other answers for details of what happens after you do this.






                                share|cite|improve this answer

















                                • 1




                                  If vectors a=[1,1,0], b=[3,2,1], and c=[1,0,2], does the mean the angle between a +b and c will be the angle between [4,3,1] and [1,0,2]?
                                  – bison72
                                  14 hours ago






                                • 1




                                  @bison72 yes, this is how vector addition works.
                                  – PaÅ­lo Ebermann
                                  9 hours ago














                                up vote
                                3
                                down vote













                                The important concept to keep in mind here is that when you add two vectors, $vec a$ and $vec b$, the result is just another vector. It has a magnitude, a direction (if its magnitude is positive), and components just like any other vector.



                                We may write the expression
                                $vec a + vec b$
                                in order to name this vector, but that does not make it into any new kind of object.
                                The formulas that apply to other vectors still apply.



                                To find the angle between $vec a + vec b$ and $vec c,$
                                take the formula you would use for the angle between any two vectors, put $vec a + vec b$
                                in the places where the first vector occurs in the formula,
                                and put $vec c$ in the places where the second vector occurs.
                                Look at other answers for details of what happens after you do this.






                                share|cite|improve this answer

















                                • 1




                                  If vectors a=[1,1,0], b=[3,2,1], and c=[1,0,2], does the mean the angle between a +b and c will be the angle between [4,3,1] and [1,0,2]?
                                  – bison72
                                  14 hours ago






                                • 1




                                  @bison72 yes, this is how vector addition works.
                                  – PaÅ­lo Ebermann
                                  9 hours ago












                                up vote
                                3
                                down vote










                                up vote
                                3
                                down vote









                                The important concept to keep in mind here is that when you add two vectors, $vec a$ and $vec b$, the result is just another vector. It has a magnitude, a direction (if its magnitude is positive), and components just like any other vector.



                                We may write the expression
                                $vec a + vec b$
                                in order to name this vector, but that does not make it into any new kind of object.
                                The formulas that apply to other vectors still apply.



                                To find the angle between $vec a + vec b$ and $vec c,$
                                take the formula you would use for the angle between any two vectors, put $vec a + vec b$
                                in the places where the first vector occurs in the formula,
                                and put $vec c$ in the places where the second vector occurs.
                                Look at other answers for details of what happens after you do this.






                                share|cite|improve this answer













                                The important concept to keep in mind here is that when you add two vectors, $vec a$ and $vec b$, the result is just another vector. It has a magnitude, a direction (if its magnitude is positive), and components just like any other vector.



                                We may write the expression
                                $vec a + vec b$
                                in order to name this vector, but that does not make it into any new kind of object.
                                The formulas that apply to other vectors still apply.



                                To find the angle between $vec a + vec b$ and $vec c,$
                                take the formula you would use for the angle between any two vectors, put $vec a + vec b$
                                in the places where the first vector occurs in the formula,
                                and put $vec c$ in the places where the second vector occurs.
                                Look at other answers for details of what happens after you do this.







                                share|cite|improve this answer













                                share|cite|improve this answer



                                share|cite|improve this answer











                                answered 17 hours ago









                                David K

                                48k339106




                                48k339106







                                • 1




                                  If vectors a=[1,1,0], b=[3,2,1], and c=[1,0,2], does the mean the angle between a +b and c will be the angle between [4,3,1] and [1,0,2]?
                                  – bison72
                                  14 hours ago






                                • 1




                                  @bison72 yes, this is how vector addition works.
                                  – PaÅ­lo Ebermann
                                  9 hours ago












                                • 1




                                  If vectors a=[1,1,0], b=[3,2,1], and c=[1,0,2], does the mean the angle between a +b and c will be the angle between [4,3,1] and [1,0,2]?
                                  – bison72
                                  14 hours ago






                                • 1




                                  @bison72 yes, this is how vector addition works.
                                  – PaÅ­lo Ebermann
                                  9 hours ago







                                1




                                1




                                If vectors a=[1,1,0], b=[3,2,1], and c=[1,0,2], does the mean the angle between a +b and c will be the angle between [4,3,1] and [1,0,2]?
                                – bison72
                                14 hours ago




                                If vectors a=[1,1,0], b=[3,2,1], and c=[1,0,2], does the mean the angle between a +b and c will be the angle between [4,3,1] and [1,0,2]?
                                – bison72
                                14 hours ago




                                1




                                1




                                @bison72 yes, this is how vector addition works.
                                – PaÅ­lo Ebermann
                                9 hours ago




                                @bison72 yes, this is how vector addition works.
                                – PaÅ­lo Ebermann
                                9 hours ago










                                up vote
                                2
                                down vote













                                Using the definition of the scalar product we get
                                $$
                                (a+b)cdot c =
                                lVert a+b rVert lVert c rVert cos angle(a+b,c)
                                $$
                                where $lVert v rVert = sqrtvcdot v$.



                                For non-zero $a+b$ and $c$ we can solve for the angle
                                $$
                                angle(a+b,c) = arccos frac(a+b)cdot clVert a+b rVert lVert c rVert
                                $$






                                share|cite|improve this answer



























                                  up vote
                                  2
                                  down vote













                                  Using the definition of the scalar product we get
                                  $$
                                  (a+b)cdot c =
                                  lVert a+b rVert lVert c rVert cos angle(a+b,c)
                                  $$
                                  where $lVert v rVert = sqrtvcdot v$.



                                  For non-zero $a+b$ and $c$ we can solve for the angle
                                  $$
                                  angle(a+b,c) = arccos frac(a+b)cdot clVert a+b rVert lVert c rVert
                                  $$






                                  share|cite|improve this answer

























                                    up vote
                                    2
                                    down vote










                                    up vote
                                    2
                                    down vote









                                    Using the definition of the scalar product we get
                                    $$
                                    (a+b)cdot c =
                                    lVert a+b rVert lVert c rVert cos angle(a+b,c)
                                    $$
                                    where $lVert v rVert = sqrtvcdot v$.



                                    For non-zero $a+b$ and $c$ we can solve for the angle
                                    $$
                                    angle(a+b,c) = arccos frac(a+b)cdot clVert a+b rVert lVert c rVert
                                    $$






                                    share|cite|improve this answer















                                    Using the definition of the scalar product we get
                                    $$
                                    (a+b)cdot c =
                                    lVert a+b rVert lVert c rVert cos angle(a+b,c)
                                    $$
                                    where $lVert v rVert = sqrtvcdot v$.



                                    For non-zero $a+b$ and $c$ we can solve for the angle
                                    $$
                                    angle(a+b,c) = arccos frac(a+b)cdot clVert a+b rVert lVert c rVert
                                    $$







                                    share|cite|improve this answer















                                    share|cite|improve this answer



                                    share|cite|improve this answer








                                    edited 18 hours ago


























                                    answered 18 hours ago









                                    mvw

                                    30.1k22150




                                    30.1k22150




















                                        up vote
                                        2
                                        down vote













                                        Just $$cos(widehatveca+vecb,vecc)=frac(veca+vecb)vecc.$$



                                        Here, $|veca+vecb|=sqrt$ and $(veca+vecb)vecc=vecavecc+vecbvecc.$






                                        share|cite|improve this answer























                                        • Why someone down voted?
                                          – Michael Rozenberg
                                          14 hours ago










                                        • Might be because someone thought it's not obvious that e.g. $ab$ is the scalar product of vectors and not simple product of their norms (because you've dropped the arrows used by the OP). But that's just my guess.
                                          – Ruslan
                                          8 hours ago











                                        • @Ruslan But there is a context. Thank you for your trying! I'll fix it.
                                          – Michael Rozenberg
                                          1 hour ago















                                        up vote
                                        2
                                        down vote













                                        Just $$cos(widehatveca+vecb,vecc)=frac(veca+vecb)vecc.$$



                                        Here, $|veca+vecb|=sqrt$ and $(veca+vecb)vecc=vecavecc+vecbvecc.$






                                        share|cite|improve this answer























                                        • Why someone down voted?
                                          – Michael Rozenberg
                                          14 hours ago










                                        • Might be because someone thought it's not obvious that e.g. $ab$ is the scalar product of vectors and not simple product of their norms (because you've dropped the arrows used by the OP). But that's just my guess.
                                          – Ruslan
                                          8 hours ago











                                        • @Ruslan But there is a context. Thank you for your trying! I'll fix it.
                                          – Michael Rozenberg
                                          1 hour ago













                                        up vote
                                        2
                                        down vote










                                        up vote
                                        2
                                        down vote









                                        Just $$cos(widehatveca+vecb,vecc)=frac(veca+vecb)vecc.$$



                                        Here, $|veca+vecb|=sqrt$ and $(veca+vecb)vecc=vecavecc+vecbvecc.$






                                        share|cite|improve this answer















                                        Just $$cos(widehatveca+vecb,vecc)=frac(veca+vecb)vecc.$$



                                        Here, $|veca+vecb|=sqrt$ and $(veca+vecb)vecc=vecavecc+vecbvecc.$







                                        share|cite|improve this answer















                                        share|cite|improve this answer



                                        share|cite|improve this answer








                                        edited 1 hour ago


























                                        answered 19 hours ago









                                        Michael Rozenberg

                                        86.9k1575178




                                        86.9k1575178











                                        • Why someone down voted?
                                          – Michael Rozenberg
                                          14 hours ago










                                        • Might be because someone thought it's not obvious that e.g. $ab$ is the scalar product of vectors and not simple product of their norms (because you've dropped the arrows used by the OP). But that's just my guess.
                                          – Ruslan
                                          8 hours ago











                                        • @Ruslan But there is a context. Thank you for your trying! I'll fix it.
                                          – Michael Rozenberg
                                          1 hour ago

















                                        • Why someone down voted?
                                          – Michael Rozenberg
                                          14 hours ago










                                        • Might be because someone thought it's not obvious that e.g. $ab$ is the scalar product of vectors and not simple product of their norms (because you've dropped the arrows used by the OP). But that's just my guess.
                                          – Ruslan
                                          8 hours ago











                                        • @Ruslan But there is a context. Thank you for your trying! I'll fix it.
                                          – Michael Rozenberg
                                          1 hour ago
















                                        Why someone down voted?
                                        – Michael Rozenberg
                                        14 hours ago




                                        Why someone down voted?
                                        – Michael Rozenberg
                                        14 hours ago












                                        Might be because someone thought it's not obvious that e.g. $ab$ is the scalar product of vectors and not simple product of their norms (because you've dropped the arrows used by the OP). But that's just my guess.
                                        – Ruslan
                                        8 hours ago





                                        Might be because someone thought it's not obvious that e.g. $ab$ is the scalar product of vectors and not simple product of their norms (because you've dropped the arrows used by the OP). But that's just my guess.
                                        – Ruslan
                                        8 hours ago













                                        @Ruslan But there is a context. Thank you for your trying! I'll fix it.
                                        – Michael Rozenberg
                                        1 hour ago





                                        @Ruslan But there is a context. Thank you for your trying! I'll fix it.
                                        – Michael Rozenberg
                                        1 hour ago













                                         

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