Scalars & Vectors

Scalar and vector quantities, combining vectors by calculation and scale drawing, resolving into components, and equilibrium of coplanar forces.

Spec Points Covered
  • Distinguish between scalar and vector quantities with examples.
  • Add two perpendicular vectors using Pythagoras and trigonometry.
  • Combine non-perpendicular vectors using the triangle and parallelogram methods.
  • Resolve a vector into two perpendicular components using sin and cos.
  • Apply vector resolutionThe smallest change in a quantity that an instrument can detect. For example, a ruler has a resolution of 1 mm. to inclined plane problems.
  • Represent coplanar forces in equilibriumAn object is in equilibrium when the resultant force on it is zero. The object is either stationary or moving at constant velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.. as a closed vector triangle.
Notes
01 A scalar has magnitude only; a vector has magnitude and direction Scalar 3.4.1.1 → 02 Vectors are represented by arrows whose length shows magnitude 3.4.1.1 → 03 Two perpendicular vectors are added using Pythagoras and trigonometry $R = \sqrt{a^2 + b^2}$ 3.4.1.1 → 04 Non-perpendicular vectors are added by scale drawing 3.4.1.1 → 05 Vector subtraction reverses the direction of the second vector 3.4.1.1 → 06 Resolving a vector splits it into two perpendicular components $F_x = F \cos \theta$ 3.4.1.1 → 07 On an inclined plane, weight resolves parallel and perpendicular to the slope $F_{\parallel} = W \sin \theta$ 3.4.1.1 → 08 Forces in equilibrium form a closed vector triangle Equilibrium 3.4.1.1 →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Resultant of two perpendicular vectors
$$R = \sqrt{a^2 + b^2}$$
  • Where:
    • $R$ = resultant magnitude
    • $a$ = first vector magnitude
    • $b$ = second vector magnitude (perpendicular)
Only valid when vectors are at 90 degrees.
Horizontal component of a vector
$$F_x = F \cos \theta$$
  • Where:
    • $F_x$ = horizontal component
    • $F$ = vector magnitude
    • $θ$ = angle to horizontal
Use when angle is measured from the horizontal.
Vertical component of a vector
$$F_y = F \sin \theta$$
  • Where:
    • $F_y$ = vertical component
    • $F$ = vector magnitude
    • $θ$ = angle to horizontal
Use when angle is measured from the horizontal.
Weight component parallel to slope
$$F_{\parallel} = W \sin \theta$$
  • Where:
    • $F_parallel$ = component parallel to slope (N)
    • $W$ = weight (N)
    • $θ$ = angle of incline
On an inclined plane, this component drives the object down the slope.
Weight component perpendicular to slope
$$F_{\perp} = W \cos \theta$$
  • Where:
    • $F_perp$ = component perpendicular to slope (N)
    • $W$ = weight (N)
    • $θ$ = angle of incline
Equals the normal reaction force if there is no motion perpendicular to slope.
Q Retrieval Practice All 10 Questions →
Q1. What is the difference between a scalar and a vector?
  • A scalar has magnitude only.
  • A vector has both magnitude and direction.
Q2. Give three examples of scalar quantities and three examples of vector quantities.
  • Scalars: distance, speed, mass (also time, energyThe capacity to do work. Measured in joules (J)., temperature).
  • Vectors: displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m)., velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹., force (also accelerationThe rate of change of velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.. A vector quantity. Measured in m s⁻²., momentum).
Q3. How do you find the resultant of two perpendicular vectors?
Use Pythagoras to find the magnitude (R = sqrt(\(a^{2}\) + \(b^{2}\))) and trigonometry (tan theta = opposite/adjacent) to find the direction.
Q4. What is the triangle method for adding vectors?
  • Link the vectors head-to-tail.
  • The resultant is the vector from the tail of the first to the head of the second.
Q5. What is the parallelogram method for adding vectors?
Link the vectors tail-to-tail, complete the parallelogram, and the resultant is the diagonal from the common tail.