I want to find the vertical or you could say perpendicular component of a → on b → Now I know that it can be found out using a → − ( a ⋅ b | b | ) b → However I wanted to know why it cannot be found out using this method I tried. What is the flaw in it? a → × b → = | a | | b | sin ⁡ θ n ^ Now n ^ should be equal to ( a → × b → ) | a → × b → | Using this I can write the expression as a → × b → = | a | | b | sin ⁡ θ × ( a → × b → ) | a → × b → | which on simplifying gives me | a → × b → | | b | = | a | sin ⁡ θ which I believe should be the perpendicular componentNow I'm probably doing something really stupid but I can't really understand where am I going wrong ?

Question

I want to find the vertical or you could say perpendicular component of             a      →        on             b      →      Now I know that it can be found out using             a      →        −      (                  a        ⋅        b                              |                b                  |                      )              b      →      However I wanted to know why it cannot be found out using this method I tried. What is the flaw in it?            a      →        ×            b      →        =      |    a      |        |    b      |    sin  ⁡      θ              n      ^       Now             n      ^       should be equal to             (                        a          →                    ×                        b          →                    )                      |                              a          →                    ×                        b          →                            |            Using this I can write the expression as             a      →        ×            b      →        =      |    a      |        |    b      |    sin  ⁡      θ    ×             (                        a          →                    ×                        b          →                    )                      |                              a          →                    ×                        b          →                            |             which on simplifying gives me                    |                              a          →                    ×                        b          →                            |                            |            b              |              =      |    a      |    sin  ⁡      θ   which I believe should be the perpendicular componentNow I&#039;m probably doing something really stupid but I can&#039;t really understand where am I going wrong ?

fairymischiefv9 · Accepted Answer

You are correct that the maginitude of the transverse component of             a      →       relative to the direction             b      ^        =            b      →            /        |              b      →            |   is given by       |              a      →        ×            b      ^            |   for             R        3  . However, this does not tell you the direction of of the transverse component.It&#039;s direction is not along             n      ^      ,             a      →        ⋅            n      ^        =  0; it&#039;s direction is along             n      ^        ×            b      ^      Of course, all of this cross-product stuff only works in 3 dimensions.