Vector Addition & Components

1. Definitions (Must Know)

• Scalar: a quantity with magnitude only.
• Vector: a quantity with magnitude and direction.
• Magnitude: the size of a quantity (always non-negative).
• Resultant: the single vector that has the same effect as adding multiple vectors.
• Component: the part of a vector in a chosen direction (often horizontal/vertical).
• Coplanar vectors: vectors that lie in the same plane.

2. Key Ideas (What Earns Marks)

• In A Level, you must be able to:
  • add and subtract coplanar vectors, and
  • represent a vector as two perpendicular components.
• Addition: place vectors head-to-tail (or use the parallelogram method). The resultant goes from the start of the first to the end of the last.
• Subtraction: vector A- vector B = vector A + (- vector B) (reverse vector B).
• Resolving: if a vector vector A makes angle θ to the +x axis (measured anticlockwise),
  • Aₓ = A cos θ
  • A_y = A sin θ
• Resultant from components:
  • Rₓ = ∑ Aₓ, R_y = ∑ A_y
  • R = square root of (Rₓ² + R_y²)
• Direction: use atan2(R_y,Rₓ) when available. If you use tan⁻¹ (|R_y/Rₓ|), identify the quadrant from the component signs before stating the final direction.

3. Detailed Explanations

A. Scalars vs vectors (don’t mix them)

You can only add/subtract vectors with vectors and scalars with scalars. If direction matters, treat it as a vector problem.

B. Graphical vector addition (quick)

Use this when the question explicitly asks for a graphical method:

1. Choose a scale.
2. Draw vectors head-to-tail.
3. Draw the resultant.

For subtraction, reverse the vector you are subtracting, then add head-to-tail.

C. Component method (most reliable)

Use this when angles are given or when accuracy matters:

1. Choose axes and define +x and +y.
2. Resolve each vector into x and y components with correct signs.
3. Add components to get Rₓ and R_y.
4. Compute magnitude and direction of the resultant.

Sanity checks:

• The resultant magnitude must satisfy |A-B| ≤ R ≤ A + B for two vectors of magnitudes A and B.
• The magnitude of a component cannot exceed the original vector magnitude.
• The signs of Rₓ and R_y must agree with the quadrant you state.

Visual example: resolving into components

If a force has magnitude F at angle θ above +x, the point (Fₓ, F_y) is where the vector ends.

Mini-example (signs):

• If a force points left, its x-component is negative.
• If a velocity points down, its y-component is negative.

D. Finding the direction (quadrant check)

The value of tan⁻¹ (R_y/Rₓ) alone is not enough to guarantee the correct direction.

Workflow:

1. Use the signs of Rₓ and R_y to decide the quadrant.
2. Compute an acute angle (e.g. tan⁻¹ (|R_y/Rₓ|)).
3. Adjust the angle to the correct quadrant.

4. Common Mistakes

• Swapping sin and cos because the angle is measured from the wrong axis.
• Forgetting negative signs for components (left/down).
• Using tan⁻¹ (R_y/Rₓ) but giving the wrong quadrant.
• Subtracting by “minus the magnitude” instead of reversing the vector direction.

5. Exam Tips

• Start with a labelled sketch (axes + angle definitions).
• Keep components in the same unit as the original quantity (e.g. N, m/s).
• If your resultant direction looks wrong, check the signs of Rₓ and R_y first.
• If the angle is measured from the +y axis, swap the trig:
  • A_y = A cos θ, Aₓ = A sin θ (then apply signs).
• Next: use component resolution in Projectile Motion and uncertainty-aware results in Uncertainty & Error Propagation.

6. Worked Examples

7. Mind Stretchers