What Is a Resultant Vector?
When two or more forces, velocities, or displacements act on an object at the same time, the combined effect can be represented by a single vector called the resultant vector. Instead of tracking every individual vector, you collapse them into one equivalent vector that tells you the net magnitude and direction.
This idea is central to mechanics, engineering, navigation, and any discipline where multiple vector quantities overlap. A pilot adjusting for crosswind, an engineer summing forces on a bridge truss, or a physics student solving a projectile problem -- all rely on finding the resultant.
The Resultant Vector Formula
Given two vectors A and B, each defined by a magnitude and an angle measured counter-clockwise from the positive x-axis, the process has three stages: decompose, sum, and recombine.
Step 1 -- Decompose into Components
Let A be the magnitude of vector A and θA its angle. Let B be the magnitude of vector B and θB its angle.
[A_{x} = A \cos \theta_{A}, \quad A_{y} = A \sin \theta_{A}]
[B_{x} = B \cos \theta_{B}, \quad B_{y} = B \sin \theta_{B}]
Step 2 -- Sum the Components
[R_{x} = A_{x} + B_{x}]
[R_{y} = A_{y} + B_{y}]
Step 3 -- Find the Resultant Magnitude and Direction
[R = \sqrt{R_{x}^{2} + R_{y}^{2}}]
[\theta_{R} = \arctan!\left(\frac{R_{y}}{R_{x}}\right)]
Where:
- R is the magnitude of the resultant vector.
- theta(R) is the direction angle of the resultant, measured counter-clockwise from the positive x-axis. When using
atan2the angle is automatically placed in the correct quadrant.
Worked Example
Suppose you have two forces acting on an object:
- Vector A: magnitude 5, angle 30 degrees
- Vector B: magnitude 8, angle 120 degrees
Decompose:
[A_{x} = 5 \cos 30^\circ = 5 \times 0.8660 = 4.3301]
[A_{y} = 5 \sin 30^\circ = 5 \times 0.5 = 2.5]
[B_{x} = 8 \cos 120^\circ = 8 \times (-0.5) = -4.0]
[B_{y} = 8 \sin 120^\circ = 8 \times 0.8660 = 6.9282]
Sum:
[R_{x} = 4.3301 + (-4.0) = 0.3301]
[R_{y} = 2.5 + 6.9282 = 9.4282]
Resultant:
[R = \sqrt{0.3301^{2} + 9.4282^{2}} = \sqrt{0.1090 + 88.8910} = \sqrt{89.0} \approx 9.4340]
[\theta_{R} = \arctan!\left(\frac{9.4282}{0.3301}\right) \approx 88.0^\circ]
The resultant force has a magnitude of approximately 9.43 and points at about 88 degrees from the positive x-axis -- nearly straight up, which makes sense because vector B's large upward component dominates.
Special Cases
Vectors in the Same Direction
When both vectors share the same angle, the resultant magnitude is simply the sum of the two magnitudes and the direction stays unchanged:
[R = A + B]
Vectors in Opposite Directions
When the vectors are 180 degrees apart, the resultant magnitude is the absolute difference and the direction aligns with the larger vector:
[R = |A - B|]
Perpendicular Vectors
When the angle between the two vectors is exactly 90 degrees, the Pythagorean theorem gives the magnitude directly:
[R = \sqrt{A^{2} + B^{2}}]
Quick Recap
- Step 1: Enter the magnitude and angle for each vector.
- Step 2: The calculator resolves each vector into x and y components using cosine and sine.
- Step 3: It sums the components to get the resultant's x and y parts.
- Step 4: It computes the resultant magnitude with the Pythagorean theorem and the direction with the inverse tangent.
Use the Resultant Vector Calculator above to combine any two vectors instantly and see the net magnitude and direction.