JEE/NEET Physics · Motion in a Plane series · Part 1 of 6 · All parts →
- A scalar is size alone (mass, time); a vector is size + direction (velocity, force)
- Vectors add tip-to-tail; perpendicular components combine by Pythagoras
- Any vector splits into components: Aₓ = A cos θ, A_y = A sin θ
- Choosing smart axes turns ugly 2-D problems into two easy 1-D problems
- Unit vectors (î, ĵ) are direction-only labels: A = Aₓî + A_yĵ
Tell someone ‘walk 5 metres’ and they’ll ask which way. Tell them ‘walk 5 metres north’ and they can start. Direction is half the information — and vectors are how physics carries it. Part 1 of the Motion in a Plane series.
- Scalars vs vectors
- Tip-to-tail addition
- Resolution: the great simplifier
- Unit vectors
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
Scalars vs Vectors
Scalars are pure amounts: 5 kg, 3 hours, 20 °C — direction is meaningless for them. Vectors carry direction: 5 m/s north-east, 20 N downward. Adding scalars is arithmetic (2 kg + 3 kg = 5 kg); adding vectors is geometry — 2 N + 3 N can be anything from 1 N to 5 N depending on their directions.
Tip-to-Tail Addition
Walk 3 m east, then 4 m north: place the second arrow’s tail at the first arrow’s tip; the sum is the single arrow from start to finish — 5 m north-east (Pythagoras). Order never matters: walk north first and you still land in the same place.
Resolution: The Great Simplifier
Any slanted vector splits into a horizontal and vertical shadow: Aₓ = A cos θ, A_y = A sin θ. Why bother? Because gravity acts only vertically and many forces act only horizontally — resolving everything onto those axes turns one twisted 2-D problem into two separate, easy 1-D problems that never talk to each other.
Unit Vectors
î and ĵ are arrows of length 1 pointing along x and y — pure direction labels. Writing A = 3î + 4ĵ says ‘three units east, four units north’: the magnitude comes back as √(3² + 4²) = 5.
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| A | the vector (arrow) — magnitude + direction | unit depends on quantity |
| θ | angle between A and the x-axis | degrees |
| Aₓ, A_y | the x- and y-shadows of A | same unit as A |
| î, ĵ | unit vectors along x, y | length exactly 1 |
Solved Examples
Aₓ = 10 cos30° = 8.66 m/s; A_y = 10 sin30° = 5 m/s.
Check: √(8.66² + 5²) = 10 ✔
Answer: 8.66 m/s and 5 m/s
Perpendicular → Pythagoras: √(36 + 64) = 10 N, at tan⁻¹(8/6) = 53° north of east.
✔
Answer: 10 N at 53° N of E
x: 3 − 5 = −2 N; y: +4 N.
Net = √(4 + 16) = √20 ≈ 4.47 N, at tan⁻¹(4/2) = 63° north of west.
Resolve first, add later — never add magnitudes blindly. ✔
Answer: √20 ≈ 4.47 N
- Adding magnitudes directly. 2 N + 3 N is 5 N ONLY if they point the same way; opposite gives 1 N, perpendicular gives 3.6 N.
- cos/sin swap. The component ALONG the angle uses cos; the one across uses sin — draw the triangle, don’t guess.
- Wrong angle. θ must be measured from the axis you’re resolving along; ‘at 30° to the vertical’ means sin and cos trade places.
- Treating vectors like scalars in equations. |A| + |B| ≠ |A + B| in general — magnitudes add only for same-direction vectors.
This Physics in Your Daily Life
- Every ‘Are you going my way?’ lift share is vector addition: two journeys combine into one resultant path — ride-sharing apps literally optimise resultants.
- Sunlight through a slanted window — the light’s energy splits into components: mostly glare off the glass when the angle is shallow: resolved components in action.
- Cutting a sandwich diagonally feels bigger because you perceive the diagonal — the resultant of the two sides; Pythagoras on your plate.
- Sailboats tack zig-zag into the wind — they resolve the wind force into forward + sideways and design the hull to kill the sideways part: vector resolution as a sport.
- Phone tilt sensors — accelerometers measure gravity’s components along the phone’s axes; your screen rotates because the phone resolved g and noticed it changed.
Push a heavy cart north while your friend pushes it east: the cart moves north-east, not ‘twice as fast’ anywhere. Forces don’t stack like money; they merge like walking directions. The sum depends on the angle — so addition itself must grow a geometry. Vectors are numbers that learned about direction.
Same 3 N and 4 N: same direction → 7 N; opposite → 1 N; at 90° → 5 N. Nothing about the forces changed except their relative direction — and the result swung across a 6-newton range. The angle between arrows is not a detail; it is half the physics.
Draw the arrows tail-to-tail; the resultant is the diagonal of the parallelogram they define. Slide the angle from 0° to 180° and watch the diagonal shrink from full sum to full difference — the whole sliding scale of vector addition in one moving picture.
Practice set (answers hidden — try first)
(NEET-level) 5 N east + 12 N north = ? magnitude:
(JEE Main-level) Components of 20 at 60° from x-axis:
(NEET-level) Two 5 N forces opposite: resultant =
(Concept) Can two vectors of magnitude 8 and 5 give resultant 2?
(JEE Main-level) A = 6î + 8ĵ: |A| =
- vector = magnitude + direction; scalar = magnitude only
- tip-to-tail addition; order doesn’t matter
- resolve: Aₓ = A cos θ, A_y = A sin θ
- perpendicular vectors → Pythagoras
- |A+B| ranges from |A|−|B| to A+B
- 🔁 vectors carry direction
- 🔁 components: cos along, sin across
- 🔁 resultant of perpendiculars: Pythagoras
- 🧠 Chant: ‘cos along, sin across’.
- 🧠 3-4-5 triangle — the exam’s favourite resultant.
- 🏠 Daily: screen rotation = phone resolving gravity.
- 🏠 Daily: sailboats tack by resolving wind.
Quick revision
- A scalar is size alone (mass, time); a vector is size + direction (velocity, force)
- Vectors add tip-to-tail; perpendicular components combine by Pythagoras
- Any vector splits into components: Aₓ = A cos θ, A_y = A sin θ
- Choosing smart axes turns ugly 2-D problems into two easy 1-D problems
- Unit vectors (î, ĵ) are direction-only labels: A = Aₓî + A_yĵ
- Resolution: the great simplifier
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