Why is the dot product of two vectors a scalar value?

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I'm having some trouble seeing why dot products are said to give scalar values. As a far as I can see, it just gives another vector that is projected onto one of the 2 original vectors. How, then, is the result a scalar quantity. Can someone please explain this to me? Thank you.






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    I'm having some trouble seeing why dot products are said to give scalar values. As a far as I can see, it just gives another vector that is projected onto one of the 2 original vectors. How, then, is the result a scalar quantity. Can someone please explain this to me? Thank you.






    linear-algebra vectors intuition







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      up vote
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      I'm having some trouble seeing why dot products are said to give scalar values. As a far as I can see, it just gives another vector that is projected onto one of the 2 original vectors. How, then, is the result a scalar quantity. Can someone please explain this to me? Thank you.






      linear-algebra vectors intuition







      share|cite|improve this question















      I'm having some trouble seeing why dot products are said to give scalar values. As a far as I can see, it just gives another vector that is projected onto one of the 2 original vectors. How, then, is the result a scalar quantity. Can someone please explain this to me? Thank you.






      linear-algebra vectors intuition




      linear-algebra vectors intuition






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      Brahadeesh

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      asked 1 hour ago









      Ethan Chan

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          4 Answers
          4






          active

          oldest

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          up vote
          4
          down vote



          accepted










          No, it doesn't give another vector. It gives the product of the length of one vector by the length of the projection of the other. This is a scalar.



          You may have been mislead by some figure.



          enter image description here



          The dot product is $|A|,|B|costheta$, not the vector $A'$.






          share|cite|improve this answer




















          • So it gives the length of A's horizontal length multiplied to the magnitude of B?
            – Ethan Chan
            52 mins ago










          • @EthanChan: horizontal is inappropriate here.
            – Yves Daoust
            52 mins ago










          • What word would be better for describing Acos(theta) in that case?
            – Ethan Chan
            51 mins ago











          • @EthanChan: as in my answer.
            – Yves Daoust
            51 mins ago

















          up vote
          3
          down vote













          $$(1,2)cdot (3,4) = 1 (3) + 2(4) = 11$$



          is a scalar.



          I think you are confusing dot product with projection.



          Suppose $u$ is a unit vector, we can project $v$ onto $u$ and its length would be $|ucdot v|$ while the projection would be $(ucdot v) u$.






          share|cite|improve this answer




















          • But in all the diagrams I see, the dot product is projected onto one of the vectors. Why is this? (eg: encrypted-tbn0.gstatic.com/…).
            – Ethan Chan
            58 mins ago







          • 2




            @EthanChan $Acdot B$ is the length of the projection of $B$ onto $A$. The length of a vector is a scalar.
            – 5xum
            55 mins ago










          • @5xum So it treats the second vector as a scalar, and the projects it onto it, to show the length of A⋅B?
            – Ethan Chan
            54 mins ago










          • The second vector is also a vector. we have $vecA cdot vecB = |vecA||vecB|cos theta$. In the event that $B$ is a unit vector, then we have $vecA cdot vecB = |vecA||cos theta$ which corresponds to the length of projections if you are familiar with trigonometry.
            – Siong Thye Goh
            51 mins ago










          • @EthanChan No, it takes two vectors and projects one onto the other, and then looks at the length of the resulting vector. That length is the scalar product.
            – 5xum
            49 mins ago

















          up vote
          1
          down vote













          A dot product, by definition, is a mapping that takes two vectors and returns a scalar.



          For example, the standard dot product on $mathbb R^n$ takes two vectors, $x=(x_1,dots, x_n)$ and $y=(y_1,dots, y_n)$, and returns their dot product, $$langle x,yrangle = sum_i=1^n x_iy_i$$ which is a real number, and thus, a scalar.






          share|cite|improve this answer



























            up vote
            0
            down vote













            The scalar product describes how an amount of one vector goes in the direction of another.



            If you moved a set of heavy books on an inclined angle, then there is a horizontal component and a vertical component to the vector descrining the force applied.



            The scalar product in this case would then describes the amount of force going in the direction of the displacemen. The work done here, is defined to be the force exerted multiplied by displacement of the books, the force here is defined to be the force in the direction of the displacement.






            share|cite|improve this answer




















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              4 Answers
              4






              active

              oldest

              votes








              4 Answers
              4






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes








              up vote
              4
              down vote



              accepted










              No, it doesn't give another vector. It gives the product of the length of one vector by the length of the projection of the other. This is a scalar.



              You may have been mislead by some figure.



              enter image description here



              The dot product is $|A|,|B|costheta$, not the vector $A'$.






              share|cite|improve this answer




















              • So it gives the length of A's horizontal length multiplied to the magnitude of B?
                – Ethan Chan
                52 mins ago










              • @EthanChan: horizontal is inappropriate here.
                – Yves Daoust
                52 mins ago










              • What word would be better for describing Acos(theta) in that case?
                – Ethan Chan
                51 mins ago











              • @EthanChan: as in my answer.
                – Yves Daoust
                51 mins ago














              up vote
              4
              down vote



              accepted










              No, it doesn't give another vector. It gives the product of the length of one vector by the length of the projection of the other. This is a scalar.



              You may have been mislead by some figure.



              enter image description here



              The dot product is $|A|,|B|costheta$, not the vector $A'$.






              share|cite|improve this answer




















              • So it gives the length of A's horizontal length multiplied to the magnitude of B?
                – Ethan Chan
                52 mins ago










              • @EthanChan: horizontal is inappropriate here.
                – Yves Daoust
                52 mins ago










              • What word would be better for describing Acos(theta) in that case?
                – Ethan Chan
                51 mins ago











              • @EthanChan: as in my answer.
                – Yves Daoust
                51 mins ago












              up vote
              4
              down vote



              accepted







              up vote
              4
              down vote



              accepted






              No, it doesn't give another vector. It gives the product of the length of one vector by the length of the projection of the other. This is a scalar.



              You may have been mislead by some figure.



              enter image description here



              The dot product is $|A|,|B|costheta$, not the vector $A'$.






              share|cite|improve this answer












              No, it doesn't give another vector. It gives the product of the length of one vector by the length of the projection of the other. This is a scalar.



              You may have been mislead by some figure.



              enter image description here



              The dot product is $|A|,|B|costheta$, not the vector $A'$.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered 54 mins ago









              Yves Daoust

              115k667210




              115k667210











              • So it gives the length of A's horizontal length multiplied to the magnitude of B?
                – Ethan Chan
                52 mins ago










              • @EthanChan: horizontal is inappropriate here.
                – Yves Daoust
                52 mins ago










              • What word would be better for describing Acos(theta) in that case?
                – Ethan Chan
                51 mins ago











              • @EthanChan: as in my answer.
                – Yves Daoust
                51 mins ago
















              • So it gives the length of A's horizontal length multiplied to the magnitude of B?
                – Ethan Chan
                52 mins ago










              • @EthanChan: horizontal is inappropriate here.
                – Yves Daoust
                52 mins ago










              • What word would be better for describing Acos(theta) in that case?
                – Ethan Chan
                51 mins ago











              • @EthanChan: as in my answer.
                – Yves Daoust
                51 mins ago















              So it gives the length of A's horizontal length multiplied to the magnitude of B?
              – Ethan Chan
              52 mins ago




              So it gives the length of A's horizontal length multiplied to the magnitude of B?
              – Ethan Chan
              52 mins ago












              @EthanChan: horizontal is inappropriate here.
              – Yves Daoust
              52 mins ago




              @EthanChan: horizontal is inappropriate here.
              – Yves Daoust
              52 mins ago












              What word would be better for describing Acos(theta) in that case?
              – Ethan Chan
              51 mins ago





              What word would be better for describing Acos(theta) in that case?
              – Ethan Chan
              51 mins ago













              @EthanChan: as in my answer.
              – Yves Daoust
              51 mins ago




              @EthanChan: as in my answer.
              – Yves Daoust
              51 mins ago










              up vote
              3
              down vote













              $$(1,2)cdot (3,4) = 1 (3) + 2(4) = 11$$



              is a scalar.



              I think you are confusing dot product with projection.



              Suppose $u$ is a unit vector, we can project $v$ onto $u$ and its length would be $|ucdot v|$ while the projection would be $(ucdot v) u$.






              share|cite|improve this answer




















              • But in all the diagrams I see, the dot product is projected onto one of the vectors. Why is this? (eg: encrypted-tbn0.gstatic.com/…).
                – Ethan Chan
                58 mins ago







              • 2




                @EthanChan $Acdot B$ is the length of the projection of $B$ onto $A$. The length of a vector is a scalar.
                – 5xum
                55 mins ago










              • @5xum So it treats the second vector as a scalar, and the projects it onto it, to show the length of A⋅B?
                – Ethan Chan
                54 mins ago










              • The second vector is also a vector. we have $vecA cdot vecB = |vecA||vecB|cos theta$. In the event that $B$ is a unit vector, then we have $vecA cdot vecB = |vecA||cos theta$ which corresponds to the length of projections if you are familiar with trigonometry.
                – Siong Thye Goh
                51 mins ago










              • @EthanChan No, it takes two vectors and projects one onto the other, and then looks at the length of the resulting vector. That length is the scalar product.
                – 5xum
                49 mins ago














              up vote
              3
              down vote













              $$(1,2)cdot (3,4) = 1 (3) + 2(4) = 11$$



              is a scalar.



              I think you are confusing dot product with projection.



              Suppose $u$ is a unit vector, we can project $v$ onto $u$ and its length would be $|ucdot v|$ while the projection would be $(ucdot v) u$.






              share|cite|improve this answer




















              • But in all the diagrams I see, the dot product is projected onto one of the vectors. Why is this? (eg: encrypted-tbn0.gstatic.com/…).
                – Ethan Chan
                58 mins ago







              • 2




                @EthanChan $Acdot B$ is the length of the projection of $B$ onto $A$. The length of a vector is a scalar.
                – 5xum
                55 mins ago










              • @5xum So it treats the second vector as a scalar, and the projects it onto it, to show the length of A⋅B?
                – Ethan Chan
                54 mins ago










              • The second vector is also a vector. we have $vecA cdot vecB = |vecA||vecB|cos theta$. In the event that $B$ is a unit vector, then we have $vecA cdot vecB = |vecA||cos theta$ which corresponds to the length of projections if you are familiar with trigonometry.
                – Siong Thye Goh
                51 mins ago










              • @EthanChan No, it takes two vectors and projects one onto the other, and then looks at the length of the resulting vector. That length is the scalar product.
                – 5xum
                49 mins ago












              up vote
              3
              down vote










              up vote
              3
              down vote









              $$(1,2)cdot (3,4) = 1 (3) + 2(4) = 11$$



              is a scalar.



              I think you are confusing dot product with projection.



              Suppose $u$ is a unit vector, we can project $v$ onto $u$ and its length would be $|ucdot v|$ while the projection would be $(ucdot v) u$.






              share|cite|improve this answer












              $$(1,2)cdot (3,4) = 1 (3) + 2(4) = 11$$



              is a scalar.



              I think you are confusing dot product with projection.



              Suppose $u$ is a unit vector, we can project $v$ onto $u$ and its length would be $|ucdot v|$ while the projection would be $(ucdot v) u$.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered 59 mins ago









              Siong Thye Goh

              83.2k1456104




              83.2k1456104











              • But in all the diagrams I see, the dot product is projected onto one of the vectors. Why is this? (eg: encrypted-tbn0.gstatic.com/…).
                – Ethan Chan
                58 mins ago







              • 2




                @EthanChan $Acdot B$ is the length of the projection of $B$ onto $A$. The length of a vector is a scalar.
                – 5xum
                55 mins ago










              • @5xum So it treats the second vector as a scalar, and the projects it onto it, to show the length of A⋅B?
                – Ethan Chan
                54 mins ago










              • The second vector is also a vector. we have $vecA cdot vecB = |vecA||vecB|cos theta$. In the event that $B$ is a unit vector, then we have $vecA cdot vecB = |vecA||cos theta$ which corresponds to the length of projections if you are familiar with trigonometry.
                – Siong Thye Goh
                51 mins ago










              • @EthanChan No, it takes two vectors and projects one onto the other, and then looks at the length of the resulting vector. That length is the scalar product.
                – 5xum
                49 mins ago
















              • But in all the diagrams I see, the dot product is projected onto one of the vectors. Why is this? (eg: encrypted-tbn0.gstatic.com/…).
                – Ethan Chan
                58 mins ago







              • 2




                @EthanChan $Acdot B$ is the length of the projection of $B$ onto $A$. The length of a vector is a scalar.
                – 5xum
                55 mins ago










              • @5xum So it treats the second vector as a scalar, and the projects it onto it, to show the length of A⋅B?
                – Ethan Chan
                54 mins ago










              • The second vector is also a vector. we have $vecA cdot vecB = |vecA||vecB|cos theta$. In the event that $B$ is a unit vector, then we have $vecA cdot vecB = |vecA||cos theta$ which corresponds to the length of projections if you are familiar with trigonometry.
                – Siong Thye Goh
                51 mins ago










              • @EthanChan No, it takes two vectors and projects one onto the other, and then looks at the length of the resulting vector. That length is the scalar product.
                – 5xum
                49 mins ago















              But in all the diagrams I see, the dot product is projected onto one of the vectors. Why is this? (eg: encrypted-tbn0.gstatic.com/…).
              – Ethan Chan
              58 mins ago





              But in all the diagrams I see, the dot product is projected onto one of the vectors. Why is this? (eg: encrypted-tbn0.gstatic.com/…).
              – Ethan Chan
              58 mins ago





              2




              2




              @EthanChan $Acdot B$ is the length of the projection of $B$ onto $A$. The length of a vector is a scalar.
              – 5xum
              55 mins ago




              @EthanChan $Acdot B$ is the length of the projection of $B$ onto $A$. The length of a vector is a scalar.
              – 5xum
              55 mins ago












              @5xum So it treats the second vector as a scalar, and the projects it onto it, to show the length of A⋅B?
              – Ethan Chan
              54 mins ago




              @5xum So it treats the second vector as a scalar, and the projects it onto it, to show the length of A⋅B?
              – Ethan Chan
              54 mins ago












              The second vector is also a vector. we have $vecA cdot vecB = |vecA||vecB|cos theta$. In the event that $B$ is a unit vector, then we have $vecA cdot vecB = |vecA||cos theta$ which corresponds to the length of projections if you are familiar with trigonometry.
              – Siong Thye Goh
              51 mins ago




              The second vector is also a vector. we have $vecA cdot vecB = |vecA||vecB|cos theta$. In the event that $B$ is a unit vector, then we have $vecA cdot vecB = |vecA||cos theta$ which corresponds to the length of projections if you are familiar with trigonometry.
              – Siong Thye Goh
              51 mins ago












              @EthanChan No, it takes two vectors and projects one onto the other, and then looks at the length of the resulting vector. That length is the scalar product.
              – 5xum
              49 mins ago




              @EthanChan No, it takes two vectors and projects one onto the other, and then looks at the length of the resulting vector. That length is the scalar product.
              – 5xum
              49 mins ago










              up vote
              1
              down vote













              A dot product, by definition, is a mapping that takes two vectors and returns a scalar.



              For example, the standard dot product on $mathbb R^n$ takes two vectors, $x=(x_1,dots, x_n)$ and $y=(y_1,dots, y_n)$, and returns their dot product, $$langle x,yrangle = sum_i=1^n x_iy_i$$ which is a real number, and thus, a scalar.






              share|cite|improve this answer
























                up vote
                1
                down vote













                A dot product, by definition, is a mapping that takes two vectors and returns a scalar.



                For example, the standard dot product on $mathbb R^n$ takes two vectors, $x=(x_1,dots, x_n)$ and $y=(y_1,dots, y_n)$, and returns their dot product, $$langle x,yrangle = sum_i=1^n x_iy_i$$ which is a real number, and thus, a scalar.






                share|cite|improve this answer






















                  up vote
                  1
                  down vote










                  up vote
                  1
                  down vote









                  A dot product, by definition, is a mapping that takes two vectors and returns a scalar.



                  For example, the standard dot product on $mathbb R^n$ takes two vectors, $x=(x_1,dots, x_n)$ and $y=(y_1,dots, y_n)$, and returns their dot product, $$langle x,yrangle = sum_i=1^n x_iy_i$$ which is a real number, and thus, a scalar.






                  share|cite|improve this answer












                  A dot product, by definition, is a mapping that takes two vectors and returns a scalar.



                  For example, the standard dot product on $mathbb R^n$ takes two vectors, $x=(x_1,dots, x_n)$ and $y=(y_1,dots, y_n)$, and returns their dot product, $$langle x,yrangle = sum_i=1^n x_iy_i$$ which is a real number, and thus, a scalar.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 57 mins ago









                  5xum

                  83.8k384149




                  83.8k384149




















                      up vote
                      0
                      down vote













                      The scalar product describes how an amount of one vector goes in the direction of another.



                      If you moved a set of heavy books on an inclined angle, then there is a horizontal component and a vertical component to the vector descrining the force applied.



                      The scalar product in this case would then describes the amount of force going in the direction of the displacemen. The work done here, is defined to be the force exerted multiplied by displacement of the books, the force here is defined to be the force in the direction of the displacement.






                      share|cite|improve this answer
























                        up vote
                        0
                        down vote













                        The scalar product describes how an amount of one vector goes in the direction of another.



                        If you moved a set of heavy books on an inclined angle, then there is a horizontal component and a vertical component to the vector descrining the force applied.



                        The scalar product in this case would then describes the amount of force going in the direction of the displacemen. The work done here, is defined to be the force exerted multiplied by displacement of the books, the force here is defined to be the force in the direction of the displacement.






                        share|cite|improve this answer






















                          up vote
                          0
                          down vote










                          up vote
                          0
                          down vote









                          The scalar product describes how an amount of one vector goes in the direction of another.



                          If you moved a set of heavy books on an inclined angle, then there is a horizontal component and a vertical component to the vector descrining the force applied.



                          The scalar product in this case would then describes the amount of force going in the direction of the displacemen. The work done here, is defined to be the force exerted multiplied by displacement of the books, the force here is defined to be the force in the direction of the displacement.






                          share|cite|improve this answer












                          The scalar product describes how an amount of one vector goes in the direction of another.



                          If you moved a set of heavy books on an inclined angle, then there is a horizontal component and a vertical component to the vector descrining the force applied.



                          The scalar product in this case would then describes the amount of force going in the direction of the displacemen. The work done here, is defined to be the force exerted multiplied by displacement of the books, the force here is defined to be the force in the direction of the displacement.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 58 mins ago









                          Kevin

                          5,148722




                          5,148722



























                               

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