I have three vectors: u → , v → and w → and this equality: u → · v → = u → · w → . Then the question is if v → = w → . At first i tried to write the equality representing components: u → · v → = u 1 · v 1 + … + u n · v n u → · w → = u 1 · w 1 + … + u n · w n Then: u 1 · v 1 + … + u n · v n = u 1 · w 1 + … + u n · w n .But I got stuck right there; I thought if c i , i ∈ [ 1 , n ] could be canceled from both terms, but don't know if that is possible somehow.

Question

I have three vectors:             u      →       ,            v      →       and             w      →       and this equality:             u      →        ·            v      →        =            u      →        ·            w      →      . Then the question is if             v      →        =            w      →      . At first i tried to write the equality representing components:            u      →        ·            v      →        =      u    1    ·      v    1    +  …  +      u    n    ·      v    n              u      →        ·            w      →        =      u    1    ·      w    1    +  …  +      u    n    ·      w    n  Then:      u    1    ·      v    1    +  …  +      u    n    ·      v    n    =      u    1    ·      w    1    +  …  +      u    n    ·      w    n    .But I got stuck right there; I thought if       c    i    ,  i  ∈  [  1  ,  n  ] could be canceled from both terms, but don&#039;t know if that is possible somehow.

Frida King · Accepted Answer

Let             u      →        =            v      →        =            0      →      and             w      →        ≠            0      →       then             u      →        ⋅            v      →        =            u      →        ⋅            w      →        =            0      →      More generally             u      →        ⋅            v      →        =            u      →        ⋅            w      →          ⟹              u      →        ⋅  (            v      →        −            w      →        )  =  0    ⟹              u      →       perpendicular to   (            v      →        −            w      →        )And the above example is true as             0      →       is perpendicular to every vector.

st3he1d0t · Accepted Answer

You can rearrange and use associativity of the dot product:            u      →        ⋅            v      →        =            u      →        ⋅            w      →          ⟹              u      →        ⋅  (            v      →        −            w      →        )  =  0Which implies that:The vector (of nonzero norm) formed by the difference             v      →        −            w      →       is orthogonal to             u      →       (also of nonzero norm)Or            u      →        is a null vectorAND/OR            v      →        −            w      →       is a null vector.The final condition implies             v      →        =            w      →      , but it is clearly not the only possibility.

I have three vectors: vec(u) , vec(v) and vec(w) and this equality: vec(u) * vec(v) =vec(u) * vec(w). Then the question is if vec(v) =vec(w).

Answered question

Answer & Explanation

New Questions in Precalculus