Resolution of vectors into two components at right angles to each other.

The idea

Resolution replaces one vector by two perpendicular vectors whose combined effect is identical. Choose axes, usually horizontal and vertical or along and perpendicular to a surface, then project the vector onto each axis using trigonometry. For a vector F at angle θ to the x-axis, the x-component is F cos θ and the y-component is F sin θ. Check with Pythagoras: Fx² + Fy² = F². For example, a 50 N force at 30° above the horizontal gives 50 cos 30° ≈ 43 N horizontally and 50 sin 30° = 25 N vertically. The components are not extra forces; they are a mathematical replacement that simplifies adding vectors and applying Newton's laws along each axis independently.

What earns the marks

• State that resolution replaces a single vector by two perpendicular components with the same combined effect as the original vector.
• Select suitable perpendicular axes for the physical situation, such as horizontal and vertical or parallel and perpendicular to a slope.
• Identify the angle between the vector and the chosen axis and apply the correct trigonometric projection, for example F cos θ along the axis and F sin θ perpendicular to it.
• Check the result using Fx² + Fy² = F² and confirm that each component is smaller than the original vector unless the vector lies along an axis.
• Use resolved components to sum forces or analyse equilibrium along each axis separately, keeping directions consistent with a stated positive sense.

Common mistakes and how to put them right

• Swapping sine and cosine because the wrong angle was identified.

  Put it right: Mark the angle between the vector and the axis, then use cosine for the component along that axis and sine for the perpendicular component.

• Treating the components as additional real forces acting alongside the original vector.

  Put it right: State clearly that the components replace the original vector for calculation purposes.

• Forgetting that a component can be negative when it points in the negative direction of the chosen axis.

  Put it right: Assign a positive sense first and include the sign in later calculations.

Exam tips

• Sketch the vector and both axes before substituting numbers; label the angle you are using so the sine or cosine choice is visible.
• Keep at least three significant figures through intermediate steps and round only the final answer to the precision justified by the data.
• After calculating components, perform a quick Pythagoras check to catch a swapped sine and cosine.
• Write the resolved components with their directions, for example 43 N horizontally and 25 N vertically, rather than giving bare numbers.