When Are Two Vectors Equal? Find the Component Conditions

Expert intro: Two vectors are equal only when each corresponding component matches exactly. It sounds obvious, but it is the place where many students lose marks by treating them as just "similar."

Detailed walkthrough

Given

$a=5i-3j,\qquad b=i+2j.$

If two vectors are equal, then their corresponding components must be the same.

So we compare:

$5i-3j = i+2j.$

This would require

$5=1 \quad\text{and}\quad -3=2,$

which is impossible.

So these vectors are not equal.

If the question is asking for the relation between them, then we can also say they are not scalar multiples either, because

$\frac{5}{1}=5 \neq \frac{-3}{2}.$

Thus,

$\boxed{a\neq b}$

💡 Pitfall guide

Don’t decide vector equality by eye just because both are written with $i$ and $j$. You must compare each component separately. Also, equality is different from parallelism: equal vectors have the same length and direction, while parallel vectors only need the same or opposite direction.

🔄 Real-world variant

If the second vector were $b=5i-3j$, then the vectors would be equal immediately. If it were $b=-5i+3j$, then they would be parallel but not equal, because the direction is reversed.

🔍 Related terms

component form, equal vectors, vector comparison