Let $\vec { v } = 3 \vec { i } + a \vec { j } + b \vec { k }$

Essay

Let $\vec { v } = 3 \vec { i } + a \vec { j } + b \vec { k }$ be a vector in space with a, b > 0.
Compute the cross product $\vec { v } \times ( 3 \vec { j } + \vec { k } )$ and then use the result and the Lagrange Multiplier method to find the values of a and b such that the magnitude of the cross product $\| \vec { v } \times ( 3 \vec { j } + \vec { k } ) \|$ is the largest with $\| \vec { v } \| = 19 .$