We have the vectors v = i + j + 2 k = ( 1 , 1 , 2 ) and u = − i − k = ( − 1 , 0 , − 1 )I want to calculate the angle between u and v in radians using the cross product.I have done the following: | v × u | = | v | ⋅ | u | ⋅ sin ⁡ θ ⇒ sin ⁡ θ = | v × u | | v | ⋅ | u | = 3 6 ⋅ 2 = 3 3 ⋅ 2 ⋅ 2 = 1 2 ⇒ θ = 2 π n + π 6 or θ = 2 π n + 5 π 6 , n ∈ Z Is everything correct?Which of the values do we choose? Or are both valid?

Question

We have the vectors   v  =  i  +  j  +  2  k  =  (  1  ,  1  ,  2  ) and   u  =  −  i  −  k  =  (  −  1  ,  0  ,  −  1  )I want to calculate the angle between u and v in radians using the cross product.I have done the following:                              |                v        ×        u                  |                =                  |                v                  |                ⋅                  |                u                  |                ⋅        sin        ⁡        θ                            ⇒        sin        ⁡        θ        =                                            |                        v            ×            u                          |                                                          |                        v                          |                        ⋅                          |                        u                          |                                      =                              3                                              6                        ⋅                          2                                      =                              3                                              3                        ⋅                          2                        ⋅                          2                                      =                  1          2                                                  ⇒        θ        =        2        π        n        +                  π          6                          or                  θ        =        2        π        n        +                              5            π                    6                ,                 n        ∈                  Z                    Is everything correct?Which of the values do we choose? Or are both valid?

devilvunga · Accepted Answer

The answer is             5      π        6   because  cos  ⁡  θ  =            −      1      −      2                      6            ⋅              2              =  −            3        2

Which is the angle of the two vectors? We have the vectors v=i+j+2k=(1,1,2) and u=−i−k=(−1,0,−1). I want to calculate the angle between u and v in radians using the cross product.

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