Q.A particle's position moves from $\vec{r}_1 = 3\hat{i}+4\hat{j}$ to $\vec{r}_2 = \hat{i}+2\hat{j}$. Calculate the displacement vector $\Delta\vec{r}$ and draw $\vec{r}_1$, $\vec{r}_2$ and $\Delta\vec{r}$ in a two dimensional Cartesian coordinate system.

Displacement Magnitude — The Straight-Line Shortcut

Imagine you walk 3 steps east, then 4 steps north. You end up at a spot that's not 7 steps away from where you started — it's only 5 steps away, diagonally. That 5 steps is your displacement magnitude.

Here's the core idea: displacement magnitude is the straight-line distance between where you began and where you ended. It doesn't care about the twists and turns of your actual path. It's the "as the crow flies" distance.

The Precise Definition

Displacement is a vector — it has both a direction and a magnitude. The magnitude of displacement (often written as $|\vec{s}|$ or simply $s$) is the length of that vector. Mathematically, if your initial position is $(x_1, y_1)$ and your final position is $(x_2, y_2)$, then:

This is just the distance formula from coordinate geometry. For the 3-step east, 4-step north example:

Why This Matters in Physics

In kinematics problems, displacement magnitude tells you the net effect of motion. When a car drives around a circular track and returns to the starting point, its displacement magnitude is zero — even though it travelled hundreds of metres. The car ended up exactly where it began.

For motion along a straight line (say, the x-axis), the displacement magnitude simplifies to:

That's just the absolute difference between final and initial positions. No square roots needed.

A Quick Check

If a particle moves from $x = 2$ m to $x = -3$ m, what's the displacement magnitude?

The displacement vector is $\vec{s} = (-3 - 2) = -5$ m. Its magnitude is $| -5 | = 5$ m. The negative sign only tells you the direction (leftwards), but the magnitude — the distance between the two points — is 5 metres.

Looking up "Displacement Magnitude: definition, formula & real-world examples" is a good habit before an exam, and it is worth knowing that Displacement Magnitude is a core, NCERT-aligned topic from the Motion in a Straight Line portion of the Class 11 Physics curriculum, and it is tested regularly in CBSE board exams as well as in JEE Main and NEET. Cross-checking this explanation against the relevant NCERT Physics chapter and solving a few past-year questions will round out your preparation.