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T.3.3 Combining and Resolving Vectors — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 8 of 22 1 quiz

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Lesson summary

T.3.3 Combining and Resolving Vectors — IB PHYSICS SL notes: Add, subtract and multiply vectors, and resolve a vector into perpendicular components using trigonometry.Vectors are combined by addition or subtraction — to give a single resultant vector (the net vector, e.g.

  • Two methods
  • The magnitude of the resultant comes from Pythagoras or trigonometry
  • To subtract a vector, reverse its direction and add
  • Resolving is the opposite of adding
  • cos θ is the adjacent side of the right-angled triangle
  • The angle must be stated from a named reference

Part 1 of 7Combining vectors

Combining vectors
Vectors are combined by addition or subtraction
  • Vectors are combined by addition or subtraction to give a single resultant vector.
    • (the net vector, e.g. net force)
Two methods
  • Two methods the triangle method and the parallelogram method.
The magnitude of the resultant comes from Pythagoras or trigonometry
  • The magnitude of the resultant comes from Pythagoras or trigonometry for perpendicular vectors.

Part 2 of 7Triangle method

Triangle method
Link the vectors head to tail.
The resultant joins the tail of the first vector to the head of the second.
1To subtract a vector, reverse its direction and add
  • To subtract a vector, reverse its direction and add c = a − b is a + (−b).
2c
c a + b is drawn from the tail of a to the head of b.

Part 3 of 7Parallelogram method

Parallelogram method
Link the vectors tail to tail.
Complete the parallelogram.
The resultant vector is the diagonal of the parallelogram.

Part 4 of 7Vector multiplication

  • The product of a scalar and a vector is always a vector.
    • Mass (scalar) times acceleration (vector) gives force: F = ma.
    • Mass (scalar) times velocity (vector) gives momentum: p = mv.

Part 5 of 7Resolving a vector

1Resolving is the opposite of adding
  • Resolving is the opposite of adding one vector is replaced by two perpendicular components with the same combined effect.
2
For a vector F at angle θ to the horizontal.
  • Horizontal component Fx = F cos θ.
  • Vertical component Fy = F sin θ.
3cos θ is the adjacent side of the right-angled triangle
  • cos θ is the adjacent side of the right-angled triangle adjacent to the angle θ is used.
4The angle must be stated from a named reference
  • The angle must be stated from a named reference the x-axis (horizontal) or vertical.

Part 6 of 7Resultant of perpendicular vectors

1Magnitude from Pythagoras
  • R = √(x2 + y2).
2Direction from trigonometry
  • Direction from trigonometry tan θ = opposite / adjacent, so θ = tan-1(y / x).
3Give both magnitude and direction in the final answer
  • Give both magnitude and direction in the final answer e.g. an angle above the horizontal.
4Use trigonometry for perpendicular vectors
  • Use trigonometry for perpendicular vectors use scale diagrams when vectors are not at right angles.

Part 7 of 7More from the specification

1Resolving velocity and mechanical power vectors
  • Resolving velocity and mechanical power vectors multiplying applied force vectors by instantaneous speed yields mechanical power output along directional components.
2Resultant range
  • Resultant range two 5 N forces combine to give anywhere from 0 N to 10 N, depending on the angle between them.
3Resultant range
  • Resultant range two 5 N forces combine to give anywhere from 0 N to 10 N, depending on the angle between them.
4Significant figures
  • Digits carrying meaningful information in a number.
5Percentage uncertainty
  • The absolute uncertainty divided by the value, times 100%.
6Absolute uncertainty
  • The range ±Δx in the same unit as the measurement.
7Percentage error
  • Difference from the accepted value as a percentage of it.
8Derived unit
  • A unit formed by multiplying or dividing SI base units.
9Percentage difference
  • Difference divided by accepted value × 100
10Scale drawing
  • Accurate drawing with a chosen scale.
11Uncertainty
  • An estimate of the range in which the true value is expected to lie.
12Gradient
  • Slope of a graph line.
13Distance
  • Total length of path travelled, a scalar.
14Variable
  • A factor that can change in an investigation.
15Length
  • Resistance is directly proportional to the length of a wire.
16Constant
  • Not changing.
17Displacement
  • Change in position.
18Proportional
  • Changing by the same factor as another quantity.
19a
− ω2x
What the symbols mean: a = acceleration (m s−2) · ω = angular frequency (rad s−1) · x = displacement from equilibrium (m); the minus sign means a always points back to the centre
20Resultant range
  • Resultant range two 5 N forces combine to give anywhere from 0 N to 10 N, depending on the angle between them.

Key terms

TermMeaning
Resultant vectorSingle vector equivalent to a combination of vectors.
ComponentPerpendicular part of a vector found by resolving.
Triangle methodAdding vectors head to tail.
Parallelogram methodAdding vectors tail to tail; resultant is the diagonal.
ResolvingSplitting a vector into perpendicular components.
Net forceResultant of all forces on an object.

Avoid these mistakes

1
Giving only a magnitude for a vector quantity — a direction is also needed.
2
Adding vectors tail to head in the wrong order — the resultant must join the first tail to the last head.
3
Using sin and cos the wrong way round — cos θ uses the side adjacent to θ.
4
Subtracting vectors without reversing the direction of the second.
5
Using Pythagoras for vectors not at right angles.

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