Key takeaways
- |v| = √(x² + y² + z²).
- Square each part first, then add, then take the square root.
- A magnitude is never negative, because squares are never negative.
- For a 2D vector, use z = 0.
In this guide
What is the magnitude of a vector?
A vector is an arrow with a direction and a size. It is written with components, like (2, 3, 6), meaning 2 steps along x, 3 along y and 6 along z. The magnitude is the straight-line length of that arrow.
Why does the magnitude matter?
Vectors describe movement, speed and force. The magnitude tells you how much: how far, how fast, how strong. The direction is handy, but often the size is the number you actually need.
A plane flying 300 km/h east while the wind pushes it 40 km/h north is not moving at 340 km/h. Its real speed is √(300² + 40²) ≈ 302.7 km/h.
The formula
| Symbol | What it means | Example |
|---|---|---|
| ‖v‖ | The magnitude (length), 0 or positive | 7 m |
| x | Component along the x-axis | 2 m |
| y | Component along the y-axis | 3 m |
| z | Component along the z-axis (0 for 2D) | 6 m |
The magnitude has the same unit as the components, for example metres or newtons. This is the Pythagorean theorem used twice: once across the floor and once up to the height.
How to find the magnitude step by step
- 1Square each component. Work out x², y² and z².
- 2Add the squares.
x² + y² + z². - 3Take the square root. The result is |v|.
Now enter your own vector in the calculator below.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Dropping negative signs wrongly.
(−3)² = 9, a positive number. A magnitude is never negative. - Forgetting the square root. x² + y² + z² is the squared length, not the length.
- Just adding the parts. 2 + 3 + 6 = 11 is the path along the axes, not the straight-line distance.
Tips and tricks
- Estimate first. The magnitude is always at least as big as the largest component (here 6) and at most the sum of all of them (here 11).
- Distance between two points? Subtract their coordinates, then find the magnitude. The distance formula calculator does this for 2D points.
- Angles between vectors use magnitudes together with the dot product.
Where you'll use it in real life
- Travel and drones: straight-line distance from the start point.
- Physics: speed from a velocity vector, or the strength of a force.
- Navigation: the real speed of a boat or plane pushed by a current or wind.
- Games and graphics: normalising direction vectors so movement speed stays the same.
Quick summary
To find the magnitude of a vector, square each component, add them, and take the square root: |v| = √(x² + y² + z²). The answer is always 0 or positive. Use the calculator above to find the length of any vector with every step shown.
Worked example
A drone flies 2 m east, 3 m north and 6 m up from its starting point. Its movement is the vector (2, 3, 6). How far is it from where it started, in a straight line?
- 01|v| = √(x² + y² + z²)
- 02|v| = √(2² + 3² + 6²)
- 03|v| = √(4 + 9 + 36)
- 04|v| = √49 = 7
These numbers are pre-filled in the calculator above.
Frequently asked questions
Can the magnitude of a vector be negative?
No. It is a length, so it is always 0 or positive. Only the zero vector (0, 0, 0) has magnitude 0.
How do I find the magnitude of a 2D vector?
Use |v| = √(x² + y²), or put z = 0 in the 3D formula. For (3, 4) the magnitude is √25 = 5.
What is a unit vector?
A vector with magnitude 1. You make one by dividing each component of a vector by its magnitude.
