If a($$\alpha$$ $$\times$$ $$\beta$$) + b($$\beta$$ $$\times$$ $$\gamma$$) + c($$\gamma$$ + $$\alpha$$) = 0, where a, b, c are non-zero scalars, then the vectors $$\alpha$$, $$\beta$$, $$\gamma$$ are

The unit vector in ZOX plane, making angles $$45^\circ $$ and $$60^\circ $$ respectively with $$\alpha = 2\widehat i + 2\widehat j - \widehat k$$ and $$\beta = \widehat j - \widehat k$$ is

Let $$\widehat \alpha $$, $$\widehat \beta $$, $$\widehat \gamma $$ be three unit vectors such that $$\widehat \alpha \, \times \,(\widehat \beta \times \widehat \gamma ) = {1 \over 2}(\widehat \beta + \widehat \gamma )$$ where $$\widehat \alpha \, \times \,(\widehat \beta \times \widehat \gamma ) = $$$$(\widehat \alpha \,.\,\widehat \gamma )\widehat \beta - (\widehat \alpha \,.\,\widehat \beta )\widehat \gamma $$. If $$\widehat \beta $$ is not parallel to $$\widehat \gamma $$, then the angle between $$\widehat \alpha $$ and $$\widehat \beta $$ is

The position vectors of the points A, B, C and D are $$3\widehat i - 2\widehat j - \widehat k$$, $$2\widehat i - 3\widehat j + 2\widehat k$$, $$5\widehat i - \widehat j + 2\widehat k$$ and $$4\widehat i - \widehat j - \lambda \widehat k$$, respectively. If the points A, B, C and D lie on a plane, the value of $$\lambda$$ is