Demonstrating Parallelism via Scalar Multiples
Consider two non-zero vectors \(\vec{a}\) and \(\vec{b}\) in \(\mathbb{R}^3\). Prove that if \(\vec{a}\) is a scalar multiple of \(\vec{b}\) (i.e., \(\vec{a} = k\vec{b}\) for some scalar \(k\)), then the lines determined by these vectors are parallel. Provide a rigorous proof of this relationship.