The value of $$\sin \left(2 \sin ^{-1} 0.8\right)$$ is equal to

A line makes the same angle '$$\alpha$$' with each of the $$x$$ and $$y$$ axes. If the angle '$$\theta$$', which it makes with the $$z$$-axis, is such that $$\sin ^2 \theta=2 \sin ^2 \alpha$$, then the angle $$\alpha$$ is

The negation of the statement pattern $$p \vee(q \rightarrow \sim r)$$ is

Let $$\bar{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$\bar{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$$ be two vectors. If $$\bar{c}$$ is a vector such that $$\bar{b} \times \bar{c}=\bar{b} \times \bar{a}$$ and $$\bar{c} \cdot \bar{a}=0$$, then $$\bar{c} \cdot \bar{b}$$ is equal to