Vector calculator (magnitude, addition and angle)
Enter two column vectors and this tool gives you the sum, difference, a scalar multiple, both magnitudes, the unit vector, the scalar (dot) product and the angle between them. Every result comes with the line of working you would be expected to show in an exam.
Enter two vectors and a scalar
a + b = (1, 9)
a − b: (5, -1) · 2a: (6, 8)
|a| = 5 · |b| = 5.3852
Unit vector of a: (0.6, 0.8)
Scalar product a·b: 14
Angle between them: 58.67°
Working
- a + b = (1, 9) — add the components separately.
- a − b = (5, -1)
- 2a = (6, 8)
- |a| = √(3² + 4²) = 5
- |b| = √(-2² + 5²) = 5.3852
- a·b = (3)(-2) + (4)(5) = 14
- cos θ = 14 ÷ (5 × 5.3852) = 0.5199, so θ = 58.67°
The method behind it
- Add or subtract vectors component by component.
- |a| = √(x² + y²); unit vector = a ÷ |a|.
- a·b = x₁x₂ + y₁y₂ and cos θ = (a·b) ÷ (|a||b|).
How to use this tool
1. Enter the components
Type the i and j components of vector a and vector b. Negative values are fine.
2. Set the scalar
Choose a multiplier k to see ka and how scaling changes the magnitude.
3. Read the results
Magnitudes use Pythagoras, and the angle comes from the scalar product formula cos θ = (a·b) ÷ (|a||b|).
Frequently asked questions
How do you find the magnitude of a vector?
Use Pythagoras on the components: for a vector with components x and y, the magnitude is √(x² + y²).
How do you add two vectors?
Add the components separately: the i components together and the j components together. Subtraction works the same way.
What is the scalar product used for?
The scalar product a·b = x₁x₂ + y₁y₂ gives the angle between two vectors through cos θ = (a·b) ÷ (|a||b|). If a·b = 0 the vectors are perpendicular.
What is a unit vector?
A vector of magnitude 1 in the same direction, found by dividing each component by the magnitude of the original vector.
Practise this properly, not just once
A solver gives you the answer; exam marks come from the working. MathsGradeUp pairs exam-board questions with step-by-step feedback and revision notes on the exact topics you keep dropping marks on.