Finding the Resultant of Two-Dimensional Vectors
Simple Explanation
The reverse of resolving a vector: given two perpendicular components, you can recombine them into a single resultant vector using the Pythagorean theorem for its magnitude and the inverse tangent for its direction.
Why Do We Need It?
After analysing horizontal and vertical motion separately, you often need the single overall speed and direction an object is actually moving — this formula puts the two components back together.
Formula
Resultant of Perpendicular Components
R = √(vₓ² + vy²), θ = tan⁻¹(vy / vₓ)
Given two perpendicular components of a vector, this formula recombines them into a single resultant vector with a magnitude and direction.
- R
- — Magnitude of the resultant vector
- vₓ
- — Horizontal (x) component
- vy
- — Vertical (y) component
- θ
- — Angle of the resultant above the x-axis
When to use it: Use to combine two perpendicular components (e.g. two forces, or a velocity's x and y parts) back into one overall vector.
Worked Example
Finding a resultant velocity
An object has a horizontal velocity component of 6 m/s and a vertical velocity component of 8 m/s. Find its resultant speed and direction.
Why Does This Work?
Because the two components are perpendicular, they form the two legs of a right triangle, with the resultant as the hypotenuse — this is exactly the situation the Pythagorean theorem and basic trigonometry are built for.
Real-Life Example
A boat crossing a river current
A boat aims straight across a river while the current pushes it downstream at the same time.
The boat's straight-across velocity and the current's downstream velocity are perpendicular components — the resultant formula gives the boat's actual speed and direction over the ground, which is neither straight across nor straight downstream.
Practice
A vector has a horizontal component of 3 m/s and a vertical component of 4 m/s. Find its resultant magnitude.
MediumCommon mistake
Forgetting that the components must be perpendicular for this formula to apply directly — combining two vectors that are not at 90° to each other needs the more general law of cosines instead.
Quick Review
- R = √(vₓ² + vy²) gives the resultant magnitude of two perpendicular components.
- θ = tan⁻¹(vy/vₓ) gives the resultant direction.
- This only works directly for components that are perpendicular (90° apart).