Let r be a vector and r be its magnitude. I want to evaluate d d t 1 r My working is d d t 1 r = d d r 1 r d r d t = − 1 r 2 d d t ( r ⋅ r ^ ) = − 1 r 2 ( r ˙ ⋅ r ^ + r ⋅ d r ^ d t ) I'm pretty sure the answer should be − 1 r 2 ( r ˙ ⋅ r ^ ) but I can't see why r ⋅ d r ^ d t = 0 . I get that r ^ is constant in magnitude, but its direction changes.

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Let r be a vector and r be its magnitude. I want to evaluate       d          d      t            1    r  My working is                              d                      d            t                                    1          r                                    =                  d                      d            r                                    1          r                                      d            r                                d            t                                                            =                              −            1                                r            2                                    d                      d            t                          (        r        ⋅                              r            ^                          )                                          =                              −            1                                r            2                          (                                            r                        ˙                          ⋅                                            r              ^                                      +                  r          ⋅                                    d                                                                    r                                    ^                                                                    d              t                                      )            I&#039;m pretty sure the answer should be             −      1              r      2        (                    r        ˙              ⋅                  r        ^              ) but I can&#039;t see why       r    ⋅                  d                                            r                        ^                                      d        t              =  0 . I get that             r      ^       is constant in magnitude, but its direction changes.

Cullen Petersen · Accepted Answer

Your proposed answer is not quite right. For clarity, let   ‖  r  ‖ be the norm of r and ⟨.,.⟩ the corresponding inner product. The we find, as   ‖  r  ‖  =      ⟨    r    ,    r    ⟩   that      d          d      t            1          ‖      r      ‖        =  −      1          ‖      r              ‖        2                  1          2      ‖      r      ‖        ⋅  2  ⟨  r  ,            r      ˙        ⟩  =  −      1          ‖      r              ‖        3              ⟨  r  ,            r      ˙        ⟩  ,which is in your (slightly foggy) notation  −      1          r      3            r    ⋅                    r        ˙              .

Let r be a vector and r be its magnitude. I want to evaluate (d)/(dt) (1)/(r)

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