Vectors describe quantities that have both size and direction, such as displacement, velocity, acceleration, and force. Many physics problems become easier when a diagonal vector is split into horizontal and vertical parts called components. These perpendicular components form a right triangle with the original vector as the hypotenuse.
Vector resolution matters because it turns angled motion or angled forces into simpler one-dimensional problems along the x and y axes.
If a vector R makes an angle θ above the positive x-axis, its horizontal component is R_x = R cos θ and its vertical component is R_y = R sin θ. The components can also be recombined using R = sqrt(R_x^2 + R_y^2) and θ = tan^-1(R_y/R_x), with attention to the correct quadrant. To add several vectors, resolve each one into x and y components, add all x components and all y components separately, then recombine the totals.
This method is used in projectile motion, force equilibrium, navigation, and any situation where directions matter.
Key Facts
- For a vector R at angle θ above the positive x-axis, R_x = R cos θ.
- For a vector R at angle θ above the positive x-axis, R_y = R sin θ.
- The magnitude of a vector from its components is R = sqrt(R_x^2 + R_y^2).
- The direction from components is θ = tan^-1(R_y/R_x), adjusted for the correct quadrant.
- Vector addition by components uses ΣR_x = R_1x + R_2x + ... and ΣR_y = R_1y + R_2y + ... .
- A negative component means the vector points partly in the negative direction of that axis.
Vocabulary
- Vector
- A quantity with both magnitude and direction, such as force, velocity, or displacement.
- Component
- One perpendicular part of a vector along a chosen axis, usually the x-axis or y-axis.
- Resolution
- The process of splitting a vector into perpendicular components.
- Resultant
- The single vector that has the same effect as two or more vectors combined.
- Quadrant
- One of the four regions of the coordinate plane that determines the signs of a vector's components.
Common Mistakes to Avoid
- Using sine for the x component and cosine for the y component without checking the angle is wrong because the adjacent side to θ uses cosine and the opposite side uses sine when θ is measured from the x-axis.
- Ignoring negative signs is wrong because components must show direction along each axis, not just size.
- Adding vector magnitudes directly is wrong when vectors point in different directions because only like components can be added directly.
- Using θ = tan^-1(R_y/R_x) without checking the quadrant is wrong because the calculator angle may not match the actual direction of the vector.
Practice Questions
- 1 A vector has magnitude 50 N and is directed 30 degrees above the positive x-axis. Find R_x and R_y.
- 2 A displacement has components R_x = -12 m and R_y = 5 m. Find the magnitude of the displacement and state the quadrant of its direction.
- 3 Two forces act on a box: 40 N east and 30 N north. Explain why the resultant is not 70 N and describe how to find its magnitude and direction.