Scalars and Vectors

Scalars and Vectors

🟢 Lite — Quick Review (1h–1d)

A scalar has magnitude only (e.g., mass = 5 kg, speed = 20 m/s, temperature = 30 °C). A vector carries both magnitude and a directional arrow (e.g., velocity = 15 m/s north, force = 10 N upward).

Must-know formulas for WAEC:

• Resultant from components: R = √(Aₓ² + Aᵧ²), where tan θ = Aᵧ/Aₓ
• Parallelogram law resultant: R = √(A² + B² + 2AB cos θ)
• Relative velocity: V_AB = V_A − V_B

High-yield exam pointers for WAEC SSCE:

• Distinguish scalars (mass, time, distance, speed, energy) from vectors (displacement, velocity, force, acceleration, weight, momentum)
• For perpendicular vectors (θ = 90°), R = √(A² + B²) directly — substitute cos 90° = 0
• Resolve vectors into horizontal (A cos θ) and vertical (A sin θ) components
• The equilibrant is numerically equal to the resultant but points in the opposite direction
• Relative velocity: observer A sees B moving at V_A minus V_B — subtract in that order

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🟡 Standard — Regular Study (2d–2mo)

What Is a Scalar?

A scalar quantity requires only magnitude to be fully specified. Mass, temperature, time, distance, speed, and energy are all scalars. Saying “the mass is 3 kg” completely describes the quantity — no direction information exists or is needed.

What Is a Vector?

A vector quantity demands both magnitude AND direction for complete description. Displacement, velocity, acceleration, force, weight, and momentum are vectors. A vector is drawn as a directed line segment — an arrow whose length represents the magnitude and whose arrowhead points in the direction.

Adding Vectors

Parallelogram Law: Place two vectors with a common tail, complete the parallelogram. The diagonal from the common tail gives the resultant R.

Triangle Law: Place vectors tip-to-tail in sequence. The resultant runs from the tail of the first vector to the tip of the last.

Component Method: Break each vector into horizontal (x) and vertical (y) components, sum each direction separately, then recombine: R = √(ΣAₓ² + ΣAᵧ²).

Quantity	Type	Example
Mass	Scalar	5 kg
Temperature	Scalar	298 K
Speed	Scalar	12 m/s
Displacement	Vector	8 m West
Force	Vector	50 N upward
Weight	Vector	9.8 N downward

Common WAEC trap: Students write “velocity = 30 m/s” without direction — this is speed, not velocity. Direction must be stated.

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🔴 Extended — Deep Study (3mo+)

Derived Formulas and Worked Calculation

The general parallelogram law R = √(A² + B² + 2AB cos θ) applies when two vectors A and B act at an angle θ between them. For two perpendicular forces of 3 N and 4 N acting at 90°: R = √(9 + 16 + 0) = 5 N. This is a classic WAEC 3-4-5 triangle — always check whether vectors are perpendicular before substituting into the simplified R = √(A² + B²).

When resolving a vector of magnitude F at angle θ to the horizontal: horizontal component Fₓ = F cos θ, vertical component Fᵧ = F sin θ. The resultant angle tan φ = Fᵧ/Fₓ.

Relative Velocity Deep Dive

For two cars A and B with velocities V_A and V_B along the same line, the velocity of A relative to B is V_AB = V_A − V_B. If A travels 25 m/s east and B travels 15 m/s east, A observes B moving 10 m/s east (V_A − V_B = 25 − 15). If B travels west at 15 m/s while A travels east at 25 m/s, A observes B approaching at 40 m/s east (V_A − (−V_B)).

Connections to Adjacent Topics

Vectors are essential for Newton’s Second Law (F = ma, where F is a vector), momentum (p = mv), and projectile motion components. Force equilibrium — when the resultant is zero — means all forces cancel to zero, which is the condition for objects at rest or moving at constant velocity.

Common Mistakes to Avoid

1. Using R = √(A² + B²) when A and B are NOT perpendicular — this formula only applies at θ = 90°
2. Direct magnitude addition: a 5 N force plus a 3 N force does not equal 8 N in general
3. Losing the negative sign when reversing direction (east vs west are opposite vectors)
4. Weight always acts vertically downward; the normal force acts perpendicular to the contact surface — they are NOT the same vector

Practice Prompts

1. Two forces of 6 N and 8 N act at 60° to each other. Find the magnitude of their resultant to 2 significant figures.
2. A boat crossing a river flows at 4 m/s east. The boat’s velocity relative to water is 3 m/s north. Find the boat’s resultant velocity and its direction relative to north.

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