Find the internal angle of the triangle between a → &amp; b → If a → = j ^ + 3 k ^ ; b → = − j ^ + 3 k ^ ; c → = 2 3 k ^ form a triangle then find the internal angle of the triangle between a → &amp; b → My approach is as followLet a → + b → = c → a → = j ^ + 3 k ^ ; b → = − j ^ + 3 k ^ ; c → = 2 3 k ^ ⇒ ( a → + b → ) 2 = ( c → ) 2 ⇒ a → 2 + b → 2 + 2 a → . b → = c → 2 ⇒ a → . b → = 2 ⇒ cos ⁡ θ = a → . b → | a → | . | b → | = 1 2 = π 3 But official answer is 2 π 3 where I am making mistake

Question

Find the internal angle of the triangle between       a    →    &amp;amp;      b    →  If       a    →    =            j      ^        +      3              k      ^        ;      b    →    =  −            j      ^        +      3              k      ^        ;      c    →    =  2      3              k      ^       form a triangle then find the internal angle of the triangle between       a    →    &amp;amp;      b    →  My approach is as followLet       a    →    +      b    →    =      c    →        a    →    =            j      ^        +      3              k      ^        ;      b    →    =  −            j      ^        +      3              k      ^        ;      c    →    =  2      3              k      ^        ⇒                    (                              a            →                    +                      b            →                          )            2        =                    (                              c            →                          )            2        ⇒                    a        →            2        +                    b        →            2        +  2      a    →    .      b    →    =                    c        →            2        ⇒      a    →    .      b    →    =  2  ⇒  cos  ⁡  θ  =                    a        →            .              b        →                            |                              a            →                          |            .              |                              b            →                          |              =      1    2    =      π    3  But official answer is             2      π        3   where I am making mistake

artirw9f · Accepted Answer

By your work  cos  ⁡  θ  =  −      2          2      ⋅      2        −      1    2    .Let             a      →        =                    A        B            →        ,             b      →        =                    B        C            →       and             c      →        =                    A        C            →        .Thus,  cos  ⁡  θ  =  cos  ⁡  ∡  A  B  C  =  cos  ⁡      (          180              ∘              −    ∡    (                  a        →              ,                  b        →              )    )    =  −      1    2    .

Aryanna Fisher · Accepted Answer

The best way to find the angle is tp find the lengths of the side vectors here they are   a  =  2  ,  b  =  2  ,  c  =      12  , so the required angle is C, then by cosine law  cos  ⁡  C  =                    a        2            +              b        2            −              c        2                    2      a      b        =            2      +      2      −      12        8    =  −      1    2      ⟹    C  =            2      π        3    .If you want to it by dot product of             a      →       and             b      →      , the angle between             a      →       and             B      →       is given by  cos  ⁡  ϕ  =                              a          →                    .                        b          →                            a      b        =      1    2      ⟹    ϕ  =      π    3    .But the internal abgke of the trangle is given by   θ  =  π  −  ϕ  =            2      π        3    .

If vec(a)=hat(j)+sqrt3 hat(k); vec(b)=−hat(j)+sqrt3 hat(k);vec(c)=2 sqrt3 hat(k) form a triangle then find the internal angle of the triangle between vec(a) & vec(b)

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