Chapter 3 / 3.5 Resolution of Vectors

Resolution of Vectors — Quick, Friendly Notes

Whenever you break a single vector into parts or put several vectors together, you’re “resolving” or “adding” them. These notes walk you through the key ideas step-by-step, using easy language and plenty of examples.

1 · Why Resolve a Vector?

Imagine sliding a heavy box: it’s easier to think of your push as a forward part and a sideways part than as one slanted shove. Mathematically, any vector $ \mathbf A $ in a plane can be written as the sum of two simpler vectors drawn along any two non-parallel directions $ \mathbf a$ and $ \mathbf b$:

$$\mathbf A = \lambda\,\mathbf a + \mu\,\mathbf b$$:contentReference[oaicite:0]{index=0}

The numbers $\lambda$ and $\mu$ tell you “how much” of each direction you need. This is like saying, “Go λ steps east and μ steps north to end up at the same spot.”

2 · Meet the Unit Vectors

To keep things tidy, we pick three unit vectors — little arrows of length 1 — that point along the $x$-, $y$-, and $z$-axes:

$\hat{\mathbf i}$ (east), $\hat{\mathbf j}$ (north), and $\hat{\mathbf k}$ (up), each with length 1. :contentReference[oaicite:1]{index=1}

3 · Breaking a Vector into $x$–$y$ Parts

For a vector that lies in the $x$-$y$ plane, draw dotted lines to the axes. Those dotted lines are the components:

$\mathbf A = A_x\,\hat{\mathbf i} + A_y\,\hat{\mathbf j}$:contentReference[oaicite:2]{index=2}

The numbers $A_x$ and $A_y$ are found with the usual right-triangle trig:

$ A_x = A\cos\theta, \qquad A_y = A\sin\theta $:contentReference[oaicite:3]{index=3}

Here, $A$ is the length (speed, force, etc.) of the vector, and $\theta$ is the angle it makes with the $x$-axis.

Need the length and angle back from the components? Easy:

$ A = \sqrt{A_x^{\,2} + A_y^{\,2}}, \qquad \tan\theta = \dfrac{A_y}{A_x} $:contentReference[oaicite:4]{index=4}

4 · Stretching into 3-D

In space we tack on a third component:

$\mathbf A = A_x\,\hat{\mathbf i} + A_y\,\hat{\mathbf j} + A_z\,\hat{\mathbf k}$ :contentReference[oaicite:5]{index=5}

The length becomes

$ A = \sqrt{A_x^{\,2} + A_y^{\,2} + A_z^{\,2}} $ :contentReference[oaicite:6]{index=6}

If $\alpha,\beta,\gamma$ are the angles with the $x$-, $y$-, and $z$-axes, then

$ A_x = A\cos\alpha,\; A_y = A\cos\beta,\; A_z = A\cos\gamma $ :contentReference[oaicite:7]{index=7}

5 · Adding Vectors the Easy Way

Write each vector in component form, then add components straight across. For two vectors $\mathbf A$ and $\mathbf B$:

$ \mathbf R = \mathbf A + \mathbf B = (A_x + B_x)\,\hat{\mathbf i} + (A_y + B_y)\,\hat{\mathbf j} $:contentReference[oaicite:8]{index=8}

So the result’s components are simply

$ R_x = A_x + B_x, \quad R_y = A_y + B_y $:contentReference[oaicite:9]{index=9}

The same trick works in 3-D: just include $z$ components. :contentReference[oaicite:10]{index=10}

6 · Worked Example — Umbrella in the Rain

A rain-drop rushes down at $35\ \text{m s}^{-1}$ while a wind pushes the drop $12\ \text{m s}^{-1}$ from east toward west. The combined (resultant) speed is

$ R = \sqrt{35^{2} + 12^{2}} \approx 37\ \text{m s}^{-1} $:contentReference[oaicite:11]{index=11}

The drop’s path tilts so that

$ \tan\theta = \dfrac{12}{35} \;\Longrightarrow\; \theta \approx 19^{\circ} $:contentReference[oaicite:12]{index=12}

A waiting student should tilt the umbrella $19^{\circ}$ toward the east from the vertical to stay dry.

High-Yield Ideas for NEET

Component method: learn $A_x = A\cos\theta,\;A_y = A\sin\theta$ for rapid splitting of any vector. :contentReference[oaicite:13]{index=13}
Unit vectors $\hat{\mathbf i},\hat{\mathbf j},\hat{\mathbf k}$: the cleanest way to write vectors and avoid sign mistakes. :contentReference[oaicite:14]{index=14}
Magnitude from components: $A = \sqrt{A_x^{\,2}+A_y^{\,2}}$ (and the 3-D version) appears often in projectile and force questions. :contentReference[oaicite:15]{index=15}
Vector addition by components: add $x$, $y$, $z$ parts separately; then rebuild the final length and angle. :contentReference[oaicite:16]{index=16}
Real-world blending of vectors: the rain-and-wind umbrella problem trains you to spot right-angle situations instantly. :contentReference[oaicite:17]{index=17}

Keep practicing — split, add, and combine vectors until the steps feel automatic. You’ve got this!