Scalars and vectors
Section: Physical Quantities and Units | Syllabus: Cambridge AS Level Physics 9702
Scalars vs Vectors All physical quantities can be classed as either scalar or vector quantities. Scalar quantities: Have magnitude (size) and a unit only. They are independent of direction. Vector quantities: Have both magnitude (with unit) and a particular direction.
Scalar Quantities Vector Quantities Mass (kg) Force / Weight / Friction (N) Length / Distance (m) Displacement (m) Time (s) Velocity (m s^-1) Speed (m s^-1) Acceleration (m s^-2) Energy / Work Done (J) Momentum (N s or kg m s^-1) Power (W) Electric Field (V m^-1) Current (A) Magnetic Flux Density (T) Pressure (Pa) & Temp (K) Gravitational Field (N kg^-1) Common Misconceptions Energy: All forms of energy (even kinetic energy associated with movement) are scalars because they have no direction.
Current: Electric current has direction in a circuit, but it is a scalar because it does not follow vector addition rules. Fields: Electric and magnetic fields are vectors (required for A Level). Adding Collinear Vectors Collinear vectors act along the same straight line.
To add them: Choose a direction as positive . Assign a negative sign to vectors acting in the opposite direction. The sum is the resultant vector (the single vector that has the same effect as all original vectors combined).
Worked Example: Aeroplane Velocity Question: An aeroplane flies with a velocity of 250 m s^-1 relative to the air. The wind blows with a velocity of 40 m s^-1. Calculate the velocity relative to the ground if flying: (a) in the same direction, (b) in the opposite direction.
Solution (a) Same direction: Velocity = 250 + 40 = 290 m s^-1 (in the direction of flight) (b) Opposite direction: Velocity = 250 + (-40) = 210 m s^-1 (in the direction of flight) Physics Tip A zero resultant force does not mean an object is stationary; it means its acceleration is zero (it could be moving at a constant velocity).
Adding Coplanar Vectors Using Scale Diagrams Coplanar vectors act in the same flat plane. When they are not in a straight line, we use the Head-to-Tail method or the Parallelogram Rule . Scale Diagram Placeholder (Visualizing vectors drawn end-to-end to find a resultant R) Worked Example: Graphical Method Question: Two forces, 120 N and 50 N, act on a mass with an angle of 120° between them.
Determine the magnitude and direction of the resultant. Graphical Solution Steps Scale: 1 cm = 20 N. Draw the 120 N force horizontally (6 cm). Use a protractor to draw the 50 N force (2.5 cm) at 120° from the first.
Draw the resultant from the start of the first to the end of the second. Resultant Magnitude: Measurement of 5.1 cm × 20 = 102 N (approx). Direction: Angle measured relative to 120 N force = ~25°. Perpendicular Vectors When two vectors act at 90° to each other, the resultant is the hypotenuse of a right-angled triangle.
We solve this using Pythagoras' Theorem and Trigonometry . Magnitude: Resultant^2 = F_1^2 + F_2^2 Direction: θ = OppositeAdjacent Worked Example: Boat in a River Question: A boat travels at 2.5 m s^-1 directly across a river.
The river flows at 0.92 m s^-1 perpendicular to the boat. Calculate the resultant velocity. Solution 1. Magnitude: v = √2.5^2 + 0.92^2 = √6.25 + 0.8464 = √7.0964 = 2.66 m s^-1 2. Direction (θ relative to path across river): θ = fraction θ = ^-1(0.368) = 20.2° away from the crossing path.
Worked Example: Resultant Components Question: An aeroplane flies North at 180 km h^-1. A side wind from the East acts at 90°. The resultant velocity is 187 km h^-1. Find the wind speed. Solution 187^2 = 180^2 + (Wind Speed)^2 Wind Speed = √187^2 - 180^2 = √34969 - 32400 = √2569 = 50.7 km h^-1 Resolving Vectors into Components Resolving a vector means splitting it into two perpendicular components (usually horizontal and vertical).
Horizontal component (along angle): F_x = F θ Vertical component (opposite angle): F_y = F θ Worked Example: Climbing Aeroplane Question: An aeroplane climbs at 10° to the horizontal with a constant velocity of 150 m s^-1.
Find the horizontal and vertical velocity components. Solution (a) Horizontal component: v_x = 150 × (10°) = 150 × 0.985 = 148 m s^-1 (b) Vertical component: v_y = 150 × (10°) = 150 × 0.174 = 26 m s^-1 Worked Example: Horse Pulling a Boat Question: A horse pulls a boat along a canal using a rope at 15° to the forward direction.
The force is 210 N. Calculate the forward component of the force. Solution F_forward = 210 × (15°) = 210 × 0.966 = 203 N Adding Non-Perpendicular Coplanar Vectors Using Components To add vectors at any angle: Resolve all vectors into x and y components.
Sum the x-components ( F_x) and y-components ( F_y). Apply Pythagoras to obtain the final resultant magnitude (R = √ F_x^2 + F_y^2). Worked Example: Resultant Pull on a Tree Question: Two ropes pull a tree.
Force A = 65 N at 20° to line X. Force B = 75 N at 15° to line X on the opposite side. Find the magnitude and direction of the resultant. Solution 1. Resolve into components (relative to line X): A_x = 65 20° = 61.1 N; A_y = 6…
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