If $$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $$ then $$\overrightarrow a + \overrightarrow b + \overrightarrow c = $$

Let $$\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j$$ and $$\overrightarrow w = \widehat i + 2\widehat j + 3\widehat k\,\,.$$ If $$\widehat n$$ is a unit vector such that $$\overrightarrow u .\widehat n = 0$$ and $$\overrightarrow v .\widehat n = 0\,\,,$$ then $$\left| {\overrightarrow w .\widehat n} \right|$$ is equal to :

The vectors $$\overrightarrow {AB} = 3\widehat i + 4\widehat k\,\,\& \,\,\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k$$ are the sides of triangle $$ABC.$$ The length of the median through $$A$$ is :

The two lines $$x=ay+b,z=cy+d$$ and $$x = a'y + b',z = c'y + d'$$ will be perpendicular, if and only if :