Vectors

The $\vec{\imath}+\vec{\jmath}+\vec{k}$ has $\sqrt{1^2+1^2+1^2}=\sqrt{3}$ of magnitude

The $4\vec{k}$ has $\sqrt{4^2}=\sqrt{16}$ of magnitude

The $3\vec{\imath}+\vec{\jmath}+\vec{k}$ has $\sqrt{3^2+1^1+1^1}=\sqrt{11}$ of magnitude

The $2\vec{\imath}+3\vec{\jmath}+\vec{k}$ has $\sqrt{2^2+3^2+1^1}=\sqrt{14}$ of magnitude

The one with the highest magnitude is $4\vec{k}$

For any vector xi +yj+zk, magnitude equals sqrt(x^2+y^2+z^2)

Calculate magnitude of each vector

Square all the vector components so 4 square is greatest