Vector Addition CalculatorFree online math calculator — instant results, no sign-up
Add two two-dimensional vectors component by component.
About the Vector Addition Calculator
The Vector Addition Calculator adds two two-dimensional vectors by combining their corresponding x and y components. Enter the four components to obtain the resulting vector as separate x and y outputs. With the default vectors (2, 3) and (4, 5), the sum is (6, 8).
- First vector x
- The first vector’s horizontal component, defaulting to 2, expressed in the same units as the other components or as a unitless value.
- First vector y
- The first vector’s vertical component, defaulting to 3, expressed in the same units as the other components or as a unitless value.
- Second vector x
- The second vector’s horizontal component, defaulting to 4, expressed in the same units as the other components or as a unitless value.
- Second vector y
- The second vector’s vertical component, defaulting to 5, expressed in the same units as the other components or as a unitless value.
Formula and method
For vectors a = (a_x, a_y) and b = (b_x, b_y), their sum is r = a + b = (a_x + b_x, a_y + b_y). Thus r_x = a_x + b_x = 2 + 4 = 6 and r_y = a_y + b_y = 3 + 5 = 8.
- aThe first two-dimensional vector.
- bThe second two-dimensional vector.
- a_xThe x component of the first vector.
- a_yThe y component of the first vector.
- b_xThe x component of the second vector.
- b_yThe y component of the second vector.
- rThe resultant vector, equal to the sum of a and b.
- r_xThe x component of the resultant vector, shown as Result x.
- r_yThe y component of the resultant vector, shown as Result y.
The calculator adds the two x components and, independently, the two y components, preserving the signs of the inputs. It then reports these sums as the components of the resultant vector. For the default inputs, the horizontal calculation is 2 + 4 = 6 and the vertical calculation is 3 + 5 = 8.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- First vector x2
- First vector y3
- Second vector x4
- Second vector y5
Results
- Result x6
- Result y8
How to use the Vector Addition Calculator
- 1Confirm that both vectors use the same coordinate axes and compatible units.
- 2Enter the first vector’s components: First vector x = 2 and First vector y = 3 for the default example.
- 3Enter the second vector’s components: Second vector x = 4 and Second vector y = 5 for the default example.
- 4Run the calculation to add the x components together and the y components together.
- 5Read Result x = 6 and Result y = 8 as the resultant vector (6, 8).
Reading your result
Result x gives the combined horizontal component; the default result of 6 points in the positive x direction. Result y gives the combined vertical component; the default result of 8 points in the positive y direction. Together, these rows describe the vector (6, 8), not two separate vector lengths. Both outputs retain the units of the inputs, or remain unitless if the inputs are unitless.
Limitations
This tool adds exactly two two-dimensional vectors supplied in component form. Both vectors must describe compatible quantities using the same coordinate axes and units; convert units or coordinate systems before entering values. It does not directly accept magnitude-and-angle inputs or third-dimensional components, and its output rows do not provide the resultant’s magnitude or direction angle.
Common questions
Why are the default outputs 6 and 8 rather than one total?+
Vector addition preserves the coordinate components: the x values give 2 + 4 = 6, while the y values give 3 + 5 = 8. Adding all four inputs into one number would discard the vector’s directional information.
Can I enter negative x or y components?+
Yes. A negative component represents a contribution along the negative direction of its axis. The calculator uses signed addition, so opposite-signed components can partially or completely cancel.
Does swapping the first and second vectors change the result?+
No. Vector addition is commutative: (2, 3) + (4, 5) and (4, 5) + (2, 3) both produce (6, 8).
Can I enter a vector’s length and angle in the x and y fields?+
No. These fields require Cartesian components, not a length and angle. First convert each vector using x = magnitude × cos(angle) and y = magnitude × sin(angle), with the angle measured from the positive x axis.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
You might also like
Vector Magnitude Calculator
Calculate the magnitude of a three-dimensional vector.
Dot Product Calculator
Calculate the dot product of two three-dimensional vectors.
Cross Product Calculator
Calculate the cross product of two three-dimensional vectors.
Angle Between Vectors Calculator
Calculate the angle between two three-dimensional vectors.
Two Linear Equations Calculator
Solve two independent linear equations in two unknowns.
Point Slope Line Calculator
Evaluate a line from a known point, slope and target x coordinate.
Results are estimates for informational purposes and are not financial, medical or legal advice.