Proving u and v is perpendicular to the sum of 2u and v

Question

Suppose u and v are such that the sum of u and v is perpendicular to u. If |v| =√(2)|u|, how can I prove that v is perpendicular to the sum of u and v .

Answer 1

If the sum of u and v is perpendicular to u then

u • (u + v) = 0

which is

(1) |u|² + u•v = 0

From the condition

|v| = √2 |u|

(1) becomes

|v|²/2 + u•v = 0.

Then the hardest part (high school algebra),

|v|² + 2u•v = 0

v•(v+2u) = 0.

So they are perpendicular.