Let $$\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k$$ and $$\overrightarrow c = {c_1}i + {c_2}j + {c_3}k$$ be three non-zero vectors such that $$\overrightarrow c $$ is a unit vector perpendicular to both the vectors $$\overrightarrow a $$ and $$\overrightarrow b .$$ If the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is $${\pi \over 6},$$ then
$${\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{b_1}} & {{b_2}} & {{b_3}} \cr {{c_1}} & {{c_2}} & {{c_3}} \cr } } \right|^2}$$ is equal to

From the point A(0, 3) on the circle $${x^2} + 4x + {(y - 3)^2} = 0$$, a chord AB is drawn and extended to a point M such that AM = 2AB. The equation of the locus of M is..........................

The expression $$2\left[ {{{\sin }^6}\left( {{\pi \over 2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \right)} \right]$$ is equal to

Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is $${1 \over 2}\,{\left| z \right|^2}$$ .