Scalars and Vectors

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Scalar Quantities — Magnitude Only:

Scalar quantities have magnitude (size/amount) but no direction. Examples: mass (kg), distance (m), speed (m/s), time (s), temperature (K), energy (J), volume (m³). Scalars are added, subtracted, multiplied, and divided using ordinary arithmetic.

Vector Quantities — Magnitude AND Direction:

Vector quantities have both magnitude and direction. Examples: displacement (m), velocity (m/s), acceleration (m/s²), force (N), momentum (kg·m/s). Vectors cannot be added with ordinary arithmetic — they must be added using vector methods.

Vector Addition Methods:

Parallelogram Law: Two vectors drawn from a common point form adjacent sides of a parallelogram. The diagonal from the common point gives the resultant. R = √(A² + B² + 2AB cos θ), where θ is the angle between A and B.

Triangle Law: Place vectors head-to-tail in sequence. The closing side (from start of first to end of last) gives the resultant.

⚡ Exam Tip: In JAMB/NECO/WAEC, questions on vector addition using the parallelogram law are very common. Always draw the parallelogram first, identify the included angle, then apply R = √(A² + B² + 2AB cos θ).

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Resolution of Vectors:

A vector can be broken into components. For a vector V making an angle θ with the horizontal:

• Horizontal component: Vx = V cos θ
• Vertical component: Vy = V sin θ

These components are perpendicular to each other and act independently. This is one of the most useful techniques in physics — converting a vector problem into an equivalent problem with perpendicular components.

Vector Multiplication:

Dot Product (Scalar Product): A · B = AB cos θ — gives a scalar. Used for work, power. Work = F · d = Fd cos θ.

Cross Product (Vector Product): A × B = AB sin θ n̂ — gives a vector perpendicular to both A and B. Used for torque, angular momentum. Direction given by right-hand grip rule.

Equilibrium of Forces:

A body is in equilibrium when the resultant of all forces acting on it is zero. ΣF = 0. For two forces in equilibrium, they must be equal in magnitude and opposite in direction (action-reaction pair). For three forces in equilibrium, they must form a closed triangle when drawn head-to-tail.

Lami’s Theorem: For a body in equilibrium under three forces: F₁/sin α = F₂/sin β = F₃/sin γ, where α, β, γ are the angles opposite to F₁, F₂, F₃ respectively.

⚡ Exam Tip: When a vector problem asks for “the angle” or “resultant,” always resolve into components first. Component method is faster and less error-prone than drawing to scale for most exam questions.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Vector Subtraction:

A − B = A + (−B), where −B is vector B reversed in direction. To find relative velocity or the difference between two vectors, always reverse the second vector and add.

Direction Cosines:

For a vector with components (Vx, Vy, Vz) and magnitude V:

• l = cos α = Vx/V (direction cosine with x-axis)
• m = cos β = Vy/V (direction cosine with y-axis)
• n = cos γ = Vz/V (direction cosine with z-axis)
• l² + m² + n² = 1 (fundamental relation)

Applications in Mechanics:

When a force acts at an angle to the horizontal on an inclined plane, resolve into components parallel and perpendicular to the plane. This simplifies the problem — the component parallel to the plane causes acceleration along the plane, while the component perpendicular only affects the normal reaction.

For a particle on an inclined plane at angle θ:

• Component down the plane: mg sin θ
• Component perpendicular to the plane: mg cos θ (this determines the frictional force if the surface is rough)

Polygon Law of Forces:

For multiple coplanar forces in equilibrium, the vectors drawn head-to-tail in sequence form a closed polygon. This is an extension of the triangle law to many vectors. Useful for checking equilibrium conditions graphically.

⚡ Exam Pattern: The most common vector question type in JAMB/NECO/WAEC involves finding the resultant of two vectors at a known angle. Always: (1) identify the magnitude of each vector, (2) identify the included angle, (3) apply R² = A² + B² + 2AB cos θ for magnitude, (4) use tan φ = (B sin θ)/(A + B cos θ) for the direction of the resultant from vector A.