If ^@ 3(u^2 + v^2 + w^2) = (u+v+w)^2, ^@ find the value of ^@u + v - 2w.^@


Answer:

^@0^@

Step by Step Explanation:
  1. Lets try to expand and solve the equation ^@ \begin{align} & 3(u^2 + v^2 + w^2) = (u+v+w)^2 .\\ \end{align} ^@
  2. ^@ \begin{align} & 3(u^2 + v^2 + w^2) = (u+v+w)^2 \\ \implies & 3(u^2 + v^2 + w^2) = u^2+v^2+w^2 + 2uv + 2vw + 2wu \\ \implies & 3(u^2 + v^2 + w^2) - (u^2+v^2+w^2) = 2uv + 2vw + 2wu \\ \implies & (u^2 + v^2 + w^2)(3-1) = 2uv + 2vw + 2wu \\ \implies & 2(u^2 + v^2 + w^2) = 2(uv + vw + wu) \\ \implies & u^2 + v^2 + w^2 = uv + vw + wu \\ \implies & u^2 + v^2 + w^2 - uv - vw - wu = 0 \\ \implies & u^2 -uv + v^2 -vw + w^2 - wu = 0 \\ \implies & u(u -v) + v(v - w) + w(w - u) = 0 \\ \end{align} ^@
  3. As can be seen from above equation, for right hand side to be zero, there are only two possibilities. Either
    (u - v) = 0, (v - w) = 0 and (w - u) = 0 ..... I
    OR
    u = 0, v = 0 and w = 0 ..... II
  4. From first possibility, we can infer u = v, v = w and w = u. This implies that u = v = w .
  5. Second Possibility states that u = 0, v = 0, w = 0. Since this also satisfies the first possibility as well, therefore, u = v = w = 0.
  6. Putting the values of u, v and w to 0 in expression u + v - 2w
    u + v - 2w = 0

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