If 3(u2+v2+w2)=(u+v+w)2, find the value of u+v−2w.
Answer:
0
- Lets try to expand and solve the equation 3(u2+v2+w2)=(u+v+w)2.
- 3(u2+v2+w2)=(u+v+w)2⟹3(u2+v2+w2)=u2+v2+w2+2uv+2vw+2wu⟹3(u2+v2+w2)−(u2+v2+w2)=2uv+2vw+2wu⟹(u2+v2+w2)(3−1)=2uv+2vw+2wu⟹2(u2+v2+w2)=2(uv+vw+wu)⟹u2+v2+w2=uv+vw+wu⟹u2+v2+w2−uv−vw−wu=0⟹u2−uv+v2−vw+w2−wu=0⟹u(u−v)+v(v−w)+w(w−u)=0
- 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 - From first possibility, we can infer u = v, v = w and w = u. This implies that u = v = w .
- Second Possibility states that u = 0, v = 0, w = 0. Since this also satisfies the first possibility as well, therefore, u = v = w = 0.
- Putting the values of u, v and w to 0 in expression u + v - 2w
u + v - 2w = 0