The position vectors of three points A, B and C respectively, where  $\vec{a} ,\vec{b} $ and $\vec{c} $  respectively, where $\vec{c} = (\cos^2 \theta)\vec{a}+(\sin^2 \theta)\vec{b}$. What is $(\vec{a} \times \vec{b}) + (\vec{b} \times \vec{c}) + (\vec{c} \times \vec{a})$ equal to?

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?