Vectors describe quantities with size and direction – displacement, velocity, force. These operations are the toolkit of mechanics, computer graphics and navigation.
How to use the Vector Calculator
- Vectors in
- Operation
- a – x
- a – y
- a – z
- b – x
- b – y
- b – z
Results update as you type, and everything is calculated in your browser.
How it works
Choose the operation; the relevant results appear. For “vector from two points”, a is the starting point and b the end point.
Tips
- Sketch the points first – a quick drawing shows whether a slope should be positive or negative.
- Keep the order of the points consistent in every formula.
- Vertical lines have an undefined slope; horizontal lines have slope zero.
- Use the same units on both axes when you measure distances or angles.
Common mistakes to avoid
- Swapping x and y coordinates.
- Subtracting the coordinates in a different order in the numerator and denominator of the slope.
- Forgetting that the distance is always positive.
Inputs explained
| Field | What to enter |
|---|---|
| Vectors in | Choose one: 2D, 3D |
| Operation | Choose one: Addition a + b, Subtraction a − b, Dot product, Angle between vectors … |
| a – x | A number |
| a – y | A number |
| a – z | A number |
| b – x | A number |
| b – y | A number |
| b – z | A number |
Example values
How the main result changes when a – x changes and everything else stays at the example values above:
| a – x | Result vector – x |
|---|---|
| 0.5 | 4.5 |
| 0.75 | 4.75 |
| 1 | 5 |
| 1.25 | 5.25 |
| 1.5 | 5.5 |
| 2 | 6 |
Formula
a ± b = (aₓ ± bₓ, a_y ± b_y, a_z ± b_z)a · b = Σ aᵢbᵢ, proj_a b = (a·b / |a|²) aWorked example
Inputs — Vectors in: 3D; Operation: Addition a + b; a – x: 1; a – y: 2; a – z: 3; b – x: 4; b – y: 0; b – z: -2.
Results — Result vector – x = 5; Result vector – y = 2; Result vector – z (3D) = 1; Length of the result = 5.47722558.
Key terms
- Vector
- A quantity with magnitude and direction.
Frequently asked questions
How do I add vectors?
Add their corresponding components.
What does the dot product tell me?
How aligned two vectors are; zero means perpendicular.
What is normalisation?
Dividing a vector by its length to get a unit vector.
How do I find the vector from P to Q?
Subtract: Q − P.
What formula does the Vector Calculator use?
a ± b = (aₓ ± bₓ, a_y ± b_y, a_z ± b_z).