If $$\mathbf{a}$$ and $$\mathbf{b}$$ are two vectors such that $$\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|} < 0$$ and $$|\mathbf{a} \cdot \mathbf{b}|=|\mathbf{a} \times \mathbf{b}|$$ then the angle between the vectors $$\mathbf{a}$$ and $$\mathbf{b}$$ is

Let $$\mathbf{a}, \mathbf{b}$$ and $$\mathbf{c}$$ be three-unit vectors and $$\mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}=0$$. If the angle between $$\mathbf{b}$$ and $$\mathbf{c}$$ is $$\frac{\pi}{3}$$. Then $$[\mathbf{a b c}]^2$$ is equal to

Let $$x$$ and $$y$$ are real numbers. If $$\mathbf{a}=(\sin x) \hat{\mathbf{i}}+(\sin y) \hat{\mathbf{j}}$$ and $$\mathbf{b}=(\cos x) \hat{\mathbf{i}}+(\cos y) \hat{\mathbf{j}}$$, then $$|\mathbf{a} \times \mathbf{b}|$$ is

A vector makes equal angles $$\alpha$$ with $$X$$ and $$Y$$-axis, and $$90 \Upsilon$$ with $$Z$$-axis. Then, $$\alpha$$ is equal to (c) 45Yand 135Y (d) $90 \mathrm{Y}$