The complete lecture — every idea comes alive in the live 3D panel on the right as you read. Scroll down; the animation keeps pace, and you can drag the arrows and forces yourself.
1 — Scalars & vectors
- Scalar — a quantity with magnitude only. distance, speed, mass, time, energy, temperature.
- Vector — a quantity with magnitude and direction. displacement, velocity, acceleration, force, momentum.
"5 metres" is a scalar; "5 metres due east" is a vector. The direction changes the physics completely.
2 — Representing a vector
A vector is drawn as a straight arrow. Its length (to a scale) is the magnitude and the arrowhead gives the direction. In print a vector is shown bold, A, or with an arrow, A⃗; its magnitude is |A| or just A.
- Equal vectors — same magnitude and direction. Negative vector — same size, opposite direction (−A).
3 — Distance vs displacement
- Distance — total path length actually travelled. A scalar.
- Displacement — the straight line from start to finish, with direction. A vector.
Exam point: run one full lap of a circular track and your distance is the whole lap, but your displacement is zero — you finished where you started.
4 — Adding vectors (head-to-tail)
To add vectors you place them head to tail; the single vector from the first tail to the last head is the resultant. This is the triangle law of vector addition.
Same linesame direction → R = A + B · opposite → R = A − B
5 — Parallelogram law & resultant
Two vectors at angle θR = √(A² + B² + 2AB cos θ)
direction: tan α = (B sin θ) / (A + B cos θ)
At right angles (θ = 90°), cos θ = 0, so R = √(A² + B²) — Pythagoras.
6 — Resolution into rectangular components
The reverse of addition: a single vector A at angle θ to the horizontal is split into two perpendicular components.
Rectangular componentsAₓ = A cos θ (horizontal)
A_y = A sin θ (vertical)
A = √(Aₓ² + A_y²) · tan θ = A_y / Aₓ
7 — Worked numericals
resultant at right angles
Two forces 3 N (east) and 4 N (north) act at a point. Find the resultant.
R = √(3² + 4²) = √25 = 5 N
direction: tan α = 4/3 → α = 53° N of E
resolution
A 50 N force acts at 30° above the horizontal. Find its components.
Aₓ = 50 cos30° = 43.3 N · A_y = 50 sin30° = 25 N
8 — Exam recap
- Scalar = magnitude only; vector = magnitude + direction.
- A vector is an arrow (length = magnitude, head = direction).
- Distance (scalar) vs displacement (vector).
- Head-to-tail / triangle law of addition.
- Parallelogram law: R = √(A² + B² + 2AB cos θ).
- Resolution: Aₓ = A cos θ, A_y = A sin θ.