The value of $$'a'$$ so that the volume of parallelopiped formed by $$\widehat i + a\widehat j + \widehat k,\widehat j + a\widehat k$$ and $$a\widehat i + \widehat k$$ becomes minimum is

If $${\overrightarrow a }$$ and $${\overrightarrow b }$$ are two unit vectors such that $${\overrightarrow a + 2\overrightarrow b }$$ and $${5\overrightarrow a - 4\overrightarrow b }$$ are perpendicular to each other then the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is

Let $$\overrightarrow V = 2\overrightarrow i + \overrightarrow j - \overrightarrow k $$ and $$\overrightarrow W = \overrightarrow i + 3\overrightarrow k .$$ If $$\overrightarrow U $$ is a unit vector, then the maximum value of the scalar triple product $$\left| {\overrightarrow U \overrightarrow V \overrightarrow W } \right|$$ is

Let $$\overrightarrow a = \overrightarrow i - \overrightarrow k ,\overrightarrow b = x\overrightarrow i + \overrightarrow j + \left( {1 - x} \right)\overrightarrow k $$ and
$$\overrightarrow c = y\overrightarrow i - x\overrightarrow j + \left( {1 + x - y} \right)\overrightarrow k .$$ Then $$\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$$ depends on