What is the area of the parallelogram having diagonals $$3\widehat i + \widehat j - 2\widehat k$$ and $$\widehat i - 3\widehat j + 4\widehat k$$ ?

Let $$\widehat a$$ and $$\widehat b$$ of two unit vectors and $$\theta$$ be the angle between them.

What is $$\cos \left( {{\theta \over 2}} \right)$$ equal to ?

Let $$\widehat a$$ and $$\widehat b$$ of two unit vectors and $$\theta$$ be the angle between them.

What is $$\sin \left( {{\theta \over 2}} \right)$$ equal to ?

Let $$\alpha = \widehat i + 2\widehat j - \widehat k$$, $$\beta = 2\widehat i - \widehat j + 3\widehat k$$ and $$\gamma = 2\widehat i + \widehat j + 6\widehat k$$ be three vectors. If $$\alpha$$ and $$\beta$$ are both perpendicular to the vector $$\delta$$ and $$\delta \,.\,\gamma = 10$$, then what is the magnitude of $$\delta$$ ?