Q.Let $\vec{a}=\hat{i}+2\hat{j}$ and $\vec{b}=2\hat{i}+\hat{j}$. Is $|\vec{a}|=|\vec{b}|$? Are the vectors $\vec{a}$ and $\vec{b}$ equal?

Concept: Vector Equality — Two vectors are equal only if they have the same magnitude and the same direction. Equal magnitudes alone are not sufficient.

Step 1: Compute $|\vec{a}|$ and $|\vec{b}|$.

So $|\vec{a}| = |\vec{b}|$.

Step 2: Check direction. The vectors are $\vec{a} = \hat{i} + 2\hat{j}$ and $\vec{b} = 2\hat{i} + \hat{j}$. Their components are not proportional: $\frac{1}{2} \neq \frac{2}{1}$. Hence they point in different directions.

Step 3: Since direction differs, $\vec{a} \neq \vec{b}$ even though magnitudes are equal.

The question asks two separate things: first, whether the magnitudes are the same, and second, whether the vectors themselves are equal. These are different checks — magnitude equality is necessary but not sufficient for vector equality.

For two vectors to be equal, they must have the same length and point in exactly the same direction. That means each corresponding component must match. Let's see what happens here.

1. Find the magnitude of $\vec{a}$. $\vec{a} = \hat{i} + 2\hat{j}$ means its components are $(1, 2)$. The magnitude is

2. Find the magnitude of $\vec{b}$. $\vec{b} = 2\hat{i} + \hat{j}$ gives components $(2, 1)$. Its magnitude is

So $|\vec{a}| = |\vec{b}| = \sqrt{5}$. The answer to the first part is yes.

3. Now check if the vectors are equal. Vector equality means $\vec{a} = \vec{b}$ only if their corresponding components are exactly the same. $\vec{a}$ has $\hat{i}$-component $1$ and $\hat{j}$-component $2$. $\vec{b}$ has $\hat{i}$-component $2$ and $\hat{j}$-component $1$. Since $1 \neq 2$, the components differ. Therefore $\vec{a} \neq \vec{b}$.

4. Why the direction is different. The direction of a vector is given by the ratio of its components. For $\vec{a}$, the slope is $2/1 = 2$; for $\vec{b}$, the slope is $1/2 = 0.5$. These are clearly different slopes, so the vectors point along different lines through the origin.

Method: Testing whether two vectors are equal (magnitude vs equality)

Use this when asked both whether magnitudes agree and whether the vectors themselves are equal — two distinct checks.

Steps

Step 1: Compare magnitudes.

Compute $|\vec{a}|$ and $|\vec{b}|$ via $\sqrt{x^2+y^2+z^2}$. Equal magnitudes answer only the length question — they are necessary but not sufficient for vector equality.

Step 2: Compare directions/components.

Vector equality demands identical corresponding components (equivalently, same direction). Check whether every component matches; if any differs, the vectors are unequal even when their lengths coincide.

Step 3: State both conclusions.

Report the magnitude comparison and the equality conclusion separately. Two vectors of equal length pointing different ways (their components not identical) are a standard example of "$|\vec{a}|=|\vec{b}|$ yet $\vec{a}\neq\vec{b}$".

Common Mistakes

Mistake 1: Concluding $\vec{a}=\vec{b}$ because $|\vec{a}|=|\vec{b}|$.

Why it's wrong: equal magnitudes are necessary but not sufficient — here both are $\sqrt5$, yet the components $(1,2)$ and $(2,1)$ differ, so the vectors point in different directions and are unequal. Correct approach: after matching magnitudes, also check that corresponding components (direction) agree.

Mistake 2: Thinking two vectors with the same components in a different order are the same vector.

Why it's wrong: $\hat{i}+2\hat{j}$ and $2\hat{i}+\hat{j}$ have swapped components and hence different directions (slopes $2$ vs $\tfrac12$). Correct approach: order matters — $a_1$ must equal $b_1$ and $a_2$ must equal $b_2$ exactly.