Vectors and Equilibrium: MDCAT Notes and MCQs
Rectangular components
- A vector A at angle θ with x-axis: Ax = A cos θ, Ay = A sin θ.
- Resultant by components: Rx = ΣAx, Ry = ΣAy; R = √(Rx² + Ry²), θ = tan−1(Ry/Rx) (check quadrant).
- Two vectors: Rmax = A + B (θ = 0°), Rmin = |A − B| (θ = 180°); perpendicular: R = √(A² + B²).
Scalar (dot) product
- A · B = AB cos θ — a scalar. Commutative (A·B = B·A).
- Maximum (AB) at θ = 0°; zero when perpendicular; negative for θ > 90°.
- î·î = ĵ·ĵ = k̂·k̂ = 1; î·ĵ = ĵ·k̂ = k̂·î = 0. A·B = AxBx + AyBy + AzBz.
- Examples: work W = F·d, power P = F·v, flux Φ = B·A, E·A.
Vector (cross) product
- A × B = AB sin θ n̂ — a vector perpendicular to the plane of A and B (right-hand rule).
- Not commutative: A × B = −(B × A). Zero for parallel/antiparallel vectors; maximum at 90°.
- î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ; î × î = 0.
- |A × B| = area of parallelogram formed by A and B (triangle = ½|A × B|).
- Examples: torque τ = r × F, angular momentum L = r × p, magnetic force F = q(v × B).
If |A·B| = |A×B|, then tan θ = 1 → θ = 45°.
PMDC learning outcomes for this unit
- Addition of Vectors (Rectangular Components): 1.1 Determine the sum of vectors using perpendicular components
- Product of Vectors (Scalar Product): 1.2 Describe scalar product of two vectors in terms of angle between them
- Product of Vectors (Vector Product): 1.3 Describe vector product of two vectors in terms of angle between them.