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Resolving vectors into components

G1.1

By the end of this lesson you should be able to

  • Apply vector analysis to resolve a vector into two perpendicular components. G1.1

Key ideas

  • A vector has magnitude and direction; a scalar has magnitude only.
  • Any 2D vector can be split (resolved) into two perpendicular components, usually horizontal (x) and vertical (y).
  • The components are the sides of a right-angled triangle; the original vector is the hypotenuse.
  • Measure θ from the positive x-axis (anticlockwise) and the signs of the components look after themselves. Otherwise use the acute angle and assign signs from the diagram.

Key formulas (QCAA formula book notation)

\[ v_x = v\cos\theta \]
\[ v_y = v\sin\theta \]
Worked example

A force of 50 N acts at 30° above the horizontal. Find its horizontal and vertical components.

  1. \(F_x = 50\cos 30^\circ = 43.3\) N
  2. \(F_y = 50\sin 30^\circ = 25.0\) N
  3. Check: \(\sqrt{43.3^2 + 25.0^2} = 50\) N ✓

Common mistakes

  • Swapping sin and cos. cos goes with the side adjacent to the angle.
  • Calculator in radians instead of degrees.
  • Forgetting negative signs for components pointing left or down.

Do

  • Read: Textbook (Oxford Physics for Queensland Units 3 & 4) section 2.1
  • Do: CYL 2.1 Q1, 3, 6, 8

Try it

GeoGebra: Vector Components (Tom Walsh). Enter a magnitude and direction: predict the x and y components first, then check. Try 150°, 225° and 300° and watch the signs.

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Watch

Further reading

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