Q.Find the values of $x$ and $y$ so that the vectors $2\hat{i} + 3\hat{j}$ and $x\hat{i} + y\hat{j}$ are equal.

Concept: Vector Equality — Two vectors are equal if and only if their corresponding components are equal.

For the vectors $2\hat{i} + 3\hat{j}$ and $x\hat{i} + y\hat{j}$ to be equal:

1. Compare the $\hat{i}$ components: $x = 2$.
2. Compare the $\hat{j}$ components: $y = 3$. …

The idea of vector equality is beautifully simple — and it’s the entire foundation of this problem. Two vectors are equal if and only if they have the same magnitude and the same direction. But when vectors are expressed in component form (using $\hat{i}$ and $\hat{j}$), this condition translates into something even more concrete: each corresponding component must match exactly.

Think of it like coordinates on a map. If I tell you that point A is at (2, 3) and point B is at (x, y), and I say the two points are the same, then you immediately know $x = 2$ and $y = 3$. Vectors in component form work the same way — the $\hat{i}$ component (the x-direction) and the $\hat{j}$ component (the y-direction) are independent of each other. There’s no cross-talk between them.

Let’s walk through it step by step.

1. Write both vectors clearly.

   $\vec{A} = 2\hat{i} + 3\hat{j}$

   $\vec{B} = x\hat{i} + y\hat{j}$

2. Apply the condition for vector equality.

   For $\vec{A} = \vec{B}$, the coefficient of $\hat{i}$ in $\vec{A}$ must equal the coefficient of $\hat{i}$ in $\vec{B}$. Similarly for $\hat{j}$. …

Method: Solving for unknowns using the equality of vectors

Use this whenever two vectors are declared equal and you must find unknown components.

Steps

Step 1: State the equality condition componentwise

Two vectors are equal iff their corresponding components are equal:

A single vector equation is really one scalar equation per direction.

Step 2: Match like components …

Common Mistakes

Mistake 1: Equating magnitudes instead of components

Why it's wrong: setting $\sqrt{x^2+y^2}=\sqrt{2^2+3^2}$ gives one equation for two unknowns and admits infinitely many wrong answers. Correct approach: equal vectors need each component equal, so $x=2$ and $y=3$.

Mistake 2: Treating "equal" as merely "parallel" …