Q.State True or False: If $|\vec{a}|=|\vec{b}|$, then necessarily it implies $\vec{a}=\pm\vec{b}$.

Concept: Vector Equality — equal magnitude does not guarantee equal or opposite direction; vectors are equal only when both magnitude and direction match.

Reasoning:

1. Two vectors $\vec{a}$ and $\vec{b}$ are equal ($\vec{a} = \vec{b}$) iff $|\vec{a}| = |\vec{b}|$ and they point in the same direction.
2. The condition $|\vec{a}| = |\vec{b}|$ only tells us their lengths are the same. They could point in any direction — same, opposite, or any other angle. …

Why This Question Matters

This is a classic trap that catches many students. The confusion comes from mixing up scalar equality (numbers) with vector equality (magnitude and direction). When you see $|\vec{a}| = |\vec{b}|$, your brain might instinctively think "they're the same size, so they must be the same vector or its opposite." But vectors live in a world with two properties: length and direction. Magnitude alone tells you nothing about direction.

Think of it this way: two people can be the same height, but that doesn't mean they are the same person or even twins. Similarly, two vectors can have identical lengths but point in completely different directions.

Step-by-Step Reasoning

1. Recall the definition of vector equality.

   Two vectors $\vec{a}$ and $\vec{b}$ are equal ($\vec{a} = \vec{b}$) if and only if they have the same magnitude and the same direction. The statement $\vec{a} = \pm \vec{b}$ means either they point exactly the same way ($+$) or exactly opposite ($-$). In both cases, the vectors are collinear (lie on the same line).

2. What does $|\vec{a}| = |\vec{b}|$ actually tell us?

   It only says the lengths are equal. That's one condition satisfied for equality, but the direction condition is completely unconstrained.

3. Construct a counterexample.

   Take $\vec{a} = 3\hat{i}$ (pointing east, length 3) and $\vec{b} = 3\hat{j}$ (pointing north, length 3).

   • $|\vec{a}| = 3$, $|\vec{b}| = 3$, so $|\vec{a}| = |\vec{b}|$ holds.
   • But $\vec{a}$ is not $\pm \vec{b}$ because $\vec{a}$ points east while $\vec{b}$ points north — they are perpendicular, not collinear. …

Method: Testing a Vector Statement with a Counterexample

Use this reasoning pattern for any true/false claim of the form "a condition on magnitudes implies a relation between the vectors."

Steps

Step 1: Separate magnitude information from direction information

A vector carries both length and direction. $|\vec{a}|=|\vec{b}|$ constrains only the lengths and says nothing about direction, so a claim that equal length forces $\vec{a}=\pm\vec{b}$ (a direction relation) should be treated as suspect.

Step 2: Attempt a counterexample …

Common Mistakes

Mistake 1: Assuming equal magnitude forces $\vec{a}=\pm\vec{b}$

Why it's wrong: magnitude fixes only length; direction is unconstrained, so equal-length vectors can point in many directions. Correct approach: $\vec{a}=\pm\vec{b}$ additionally needs collinearity, which $|\vec{a}|=|\vec{b}|$ does not give.

Mistake 2: Reasoning as if vectors live on a line (1D) …