If x ^ , y ^ and z ^ are three unit vectors in three-dimensional space, then the minimum value of | x ^ + y ^ | 2 + | y ^ + z ^ | 2 + | z ^ + x ^ | 2 is:My Attempt : Let θ 1 be angle between x ^ and y ^ , θ 2 be angle between y ^ and z ^ and θ 3 be angle between z ^ and x ^ ∴ θ 1 , θ 2 and θ 3 are the angles between any 2 edges of a tetrahedron from a single vertex. ∴ | x ^ + y ^ | 2 + | y ^ + z ^ | 2 + | z ^ + x ^ | 2 = ( 2 + 2 cos ⁡ θ 1 ) + ( 2 + 2 cos ⁡ θ 2 ) + ( 2 + 2 cos ⁡ θ 3 ) = 6 + 2 ( cos ⁡ θ 1 + cos ⁡ θ 2 + cos ⁡ θ 3 )Can anyone please tell me what to do next?

Question

If             x      ^      ,             y      ^       and             z      ^       are three unit vectors in three-dimensional space, then the minimum value of       |              x      ^        +            y      ^                  |        2    +      |              y      ^        +            z      ^                  |        2    +      |              z      ^        +            x      ^                  |        2   is:My Attempt : Let       θ    1   be angle between             x      ^       and             y      ^      ,       θ    2   be angle between             y      ^       and             z      ^       and       θ    3   be angle between             z      ^       and             x      ^        ∴      θ    1    ,      θ    2   and       θ    3   are the angles between any 2 edges of a tetrahedron from a single vertex.  ∴      |              x      ^        +            y      ^                  |        2    +      |              y      ^        +            z      ^                  |        2    +      |              z      ^        +            x      ^                  |        2    =  (  2  +  2  cos  ⁡      θ    1    )  +  (  2  +  2  cos  ⁡      θ    2    )  +  (  2  +  2  cos  ⁡      θ    3    )  =  6  +  2  (  cos  ⁡      θ    1    +  cos  ⁡      θ    2    +  cos  ⁡      θ    3    )Can anyone please tell me what to do next?

mignonechatte00f · Accepted Answer

You can rotate the picture and assume             x      ^        =  (  cos  ⁡  θ  ,  sin  ⁡  θ  ,  0  ) and             y      ^        =  (  cos  ⁡  θ  ,  −  sin  ⁡  θ  ,  0  ) for some   0  ≤  θ  ≤            π      2      . Writing   z  =  (      z    1    ,      z    2    ,      z    3    ) you have            x      ^        ⋅            y      ^        +            x      ^        ⋅            z      ^        +            y      ^        ⋅            z      ^        =      cos    2    ⁡  θ  −      sin    2    ⁡  θ  +  2      z    1    cos  ⁡  θThis is minimized for   z  =  (  −  1  ,  0  ,  0  ), in which case we have            x      ^        ⋅            y      ^        +            x      ^        ⋅            z      ^        +            y      ^        ⋅            z      ^        =      cos    2    ⁡  θ  −      sin    2    ⁡  θ  −  2  cos  ⁡  θ  =  2      cos    2    ⁡  θ  −  2  cos  ⁡  θ  −  1The function   2      x    2    −  2  x  −  1 has its minimum at   x  =            1      2      , so   2      cos    2    ⁡  θ  −  2  cos  ⁡  θ  −  1 is minimized when   θ  =            π      3      Thus we have             x      ^        =  (            1      2        ,                    3            2        ,  0  ),             y      ^        =  (            1      2        ,  −                    3            2        ,  0  ), and   z  =  (  −  1  ,  0  ,  0  ), corresponding to             x      ^        ⋅            y      ^        +            x      ^        ⋅            z      ^        +            y      ^        ⋅            z      ^        =  2  (            1      2            )    2    −  2            1      2        +  1  =  −            3      2      , so that      |              x      ^        +            y      ^                  |        2    +      |              y      ^        +            z      ^                  |        2    +      |              z      ^        +            x      ^                  |        2    ≥  6  −  3=3

Howard Nelson · Accepted Answer

∑      |              x      →        +            y      →                  |        2    =  6  +  2  ∑            x      →        ⋅            y      →        ≥  6  −  3  ≥  3Above i am using      |              x      →        +            y      →        +            z      →                  |        2    ≥  0  ⟹  3  +  2  ∑            x      →        ⋅            y      →        ≥  0  ⟹  ∑            x      →        ⋅            y      →        ≥  −      3    2

Find minimum value of (cos theta_1+cos theta_2+cos theta_3)

Answered question

Answer & Explanation

New Questions in Precalculus