Given the vectors $\mathbf{a} = (2,-1)$ and $\mathbf{b} = (4,m)$

Expert intro: For parallel vectors, corresponding components are proportional. That gives a simple equation for the unknown component.

Detailed walkthrough

Since $\mathbf{a}=(2,-1)$ and $\mathbf{b}=(4,m)$ are parallel, there exists a scalar $k$ such that

$(4,m)=k(2,-1).$

Compare the first components:

$4=2k \Rightarrow k=2.$

Then compare the second components:

$m=2(-1)=-2.$

So the value of the real number is

$\boxed{-2}$

💡 Pitfall guide

A common error is to set the component ratios in the wrong order or to forget that the same scalar must work for both coordinates. Another mistake is using perpendicular logic instead of parallel logic.

🔄 Real-world variant

If the vectors were perpendicular instead of parallel, you would use the dot product: $2\cdot4+(-1)m=0$, which would give a different value of $m$.

🔍 Related terms

parallel vectors, scalar multiple, proportional components