Vectors and Equilibrium
Vectors are the mathematical foundation of mechanics. The PMDC MDCAT 2026 syllabus lists exactly three subtopics here: adding vectors via rectangular components, the scalar (dot) product, and the vector (cross) product. Expect 1-2 MCQs from this chapter, often as part of a hybrid mechanics problem.
Addition of Vectors (Rectangular Components)
A vector A in 2-D can be resolved into perpendicular (rectangular) components along x and y axes:
A = Axî + Ayĵ
where î, ĵ, k̂ are the unit vectors along x, y, z. The components are:
- Ax = A·cosθ
- Ay = A·sinθ
Conversely, given the components, the magnitude and direction of A are:
- |A| = √(Ax² + Ay²)
- tanθ = Ay/Ax
Adding two vectors
To add A and B by components: split each into x and y components, add the components separately, then re-combine:
- Rx = Ax + Bx
- Ry = Ay + By
- |R| = √(Rx² + Ry²)
- tanφ = Ry/Rx
Scalar Product
The scalar (dot) product of two vectors A and B is a scalar quantity defined by:
A·B = |A||B|cosθ
where θ is the angle between A and B. In component form:
A·B = AxBx + AyBy + AzBz
Properties
- Commutative: A·B = B·A.
- Distributive: A·(B + C) = A·B + A·C.
- Self-product: A·A = |A|².
- If A ⊥ B then A·B = 0.
- If A ∥ B then A·B = |A||B|.
- Unit vectors: î·î = ĵ·ĵ = k̂·k̂ = 1; î·ĵ = ĵ·k̂ = k̂·î = 0.
Physical examples
- Work done by a constant force: W = F·d·cosθ = F·d (dot product).
- Power: P = F·v.
- Magnetic flux: Φ = B·A.
Vector Product
The vector (cross) product of A and B is a vector defined by:
A × B = |A||B|sinθ·n̂
where n̂ is a unit vector perpendicular to the plane containing A and B, with direction given by the right-hand rule: curl the fingers of the right hand from A to B; the thumb points in the direction of A × B.
Properties
- Anti-commutative: A × B = −(B × A).
- Distributive: A × (B + C) = A × B + A × C.
- Self-product: A × A = 0.
- If A ∥ B then A × B = 0.
- If A ⊥ B then |A × B| = |A||B|.
- Unit vectors (cyclic order): î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ. Reverse order gives a minus sign.
Physical examples
- Torque: τ = r × F.
- Angular momentum: L = r × p.
- Magnetic force on a moving charge: F = qv × B.
- Magnitude of A × B = area of the parallelogram with sides A and B.
Determinant form
For A = Axî + Ayĵ + Azk̂ and B = Bxî + Byĵ + Bzk̂:
A × B = (AyBz − AzBy)î + (AzBx − AxBz)ĵ + (AxBy − AyBx)k̂
| Property | Scalar (dot) product A · B | Vector (cross) product A × B |
|---|---|---|
| Result | Scalar | Vector |
| Definition | |A| |B| cosθ | |A| |B| sinθ · n̂ |
| Maximum when | θ = 0° → |A||B| | θ = 90° → |A||B| |
| Zero when | θ = 90° (perpendicular) | θ = 0° or 180° (parallel / antiparallel) |
| Commutative? | Yes — A · B = B · A | No — A × B = −(B × A) |
| Self-product | A · A = |A|² | A × A = 0 |
| Geometric meaning | Projection of one onto the other | Area of parallelogram with sides A, B |
| Physics examples | Work W = F · d, Power P = F · v, Flux Φ = B · A | Torque τ = r × F, Angular momentum L = r × p, F = qv × B |
Worked MCQs
Five MCQs that capture the high-yield testing patterns for this chapter.
Q1. The magnitude of the resultant of two perpendicular vectors of magnitudes 3 and 4 is:
|R| = √(3² + 4²) = √25 = 5. Classic 3-4-5 triangle.
Q2. If A·B = 0 and neither A nor B is zero, then the angle between them is:
A·B = |A||B|cosθ. Zero requires cosθ = 0, i.e. θ = 90° (perpendicular vectors).
Q3. The vector product î × ĵ equals:
Cyclic order in right-handed system: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ. Reverse order picks up a minus sign.
Q4. The torque produced by a force F applied at a position vector r relative to a pivot is:
Torque is a vector quantity defined as τ = r × F (cross product). Note that F × r would give the wrong sign.
Q5. If A = 2î + 3ĵ and B = 4î − ĵ, then A·B equals:
A·B = (2)(4) + (3)(−1) = 8 − 3 = 5.
Quick Recap
- A = Axî + Ayĵ; |A| = √(Ax² + Ay²); tanθ = Ay/Ax.
- Add by components: Rx = ΣAx, Ry = ΣAy.
- A·B = |A||B|cosθ = AxBx + AyBy; commutative.
- A × B = |A||B|sinθ·n̂; anti-commutative.
- î×ĵ=k̂, ĵ×k̂=î, k̂×î=ĵ (cyclic).
- Work = F·d (dot); Torque = r × F (cross).