(a) If a and b are two vectors such that | a + b |=| b |, then prove that ( a +2 b ) is perpendicular to a.

If $\displaystyle \overrightarrow{\mathrm{a}}$ and $\displaystyle \overrightarrow{\mathrm{b}}$ are two vectors such that $\displaystyle |\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{b}}|$, then prove that $\displaystyle (\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{~b}})$ is perpendicular to $\displaystyle \overrightarrow{\mathrm{a}}$.

If $\displaystyle \overrightarrow{\mathrm{a}}$ and $\displaystyle \overrightarrow{\mathrm{b}}$ are unit vectors and $\displaystyle \theta$ is the angle between them, then prove that $\displaystyle \sin \frac{\theta}{2}=\frac{1}{2}|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|$.