Let $\vec{a},\vec{b} ,(\vec{a}\times\vec{b})$ be unit vectors. What is $\vec{a}.\vec{b}$$\stackrel{equal\; to?}{}$

What is 3α + 2β equal to if (2î + 6ĵ + 27k̂) × (î + αĵ + βk̂) is a null vector?

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?