For what value of the angle between the vectors $\rm \vec a\ and \ \vec b$ is the quantity $\rm |\vec a\times \vec b|+\sqrt3|\vec a.\vec b|$ maximum? 

Let θ be the angle between two unit vectors $\rm \vec a\ and\ \vec b. \ if\ \vec a+2\vec b$ is perpendicular to $\rm 5\vec a-4\vec b$ then what is cos θ + cos 2θ equal to?

Let ABCDEF be a regular hexagon. If $\rm \vec{AD}=m \vec {BC}\ and\ \vec {CF}=n\vec {AB}$ then what is mn equal to 

The vectors $\rm \vec a, \vec b\ and\ \vec c$ are of the same length. If taken pairwise they form equal angles. If $\rm \vec a=̂ i+̂ j\ and \ \vec b=̂ j+̂ k,$ then what can $\vec c$ be equal to? 

I. î + k̂ 

II. $\rm \frac{-\hat i+4\hat j-\hat k}{3}$

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