The sine of the angle between vectors $$a = 2\widehat i - 6\widehat j - 3\widehat k$$ and $$b = 4\widehat i + 3\widehat j - \widehat k$$ is

What is the value of $$\lambda$$ for which the vectors $$3\widehat i + 4\widehat j - \widehat k$$ and $$ - 2\widehat i + \lambda \widehat j + 10\widehat k$$ are perpendicular?

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?