Vector Magnitude Calculator
Enter 3D vector components to compute magnitude and azimuth. a=(3,4,0) → |a|=5.
Input Data
Results
At a glance:The magnitude (norm, length) of a vector is its length, denoted |a| or a. For a 3D vector |a|=√(ax²+ay²+az²); for 2D |a|=√(ax²+ay²) (Pythagorean theorem). Azimuth = atan2(ay, ax) describes the direction in the xy-plane. Physical meaning: magnitude is a scalar representing the numerical value of a quantity (force magnitude, speed, displacement distance). Components are the projections of the vector onto the coordinate axes. In polar coordinates a vector is magnitude + direction (azimuth, elevation). Applications: (1) force magnitude (resultant force); (2) speed (magnitude of velocity); (3) electric/magnetic field strength; (4) displacement distance; (5) 3D graphics, navigation, robot localization.
Formula
3D magnitude: |a| = √(ax² + ay² + az²)
2D magnitude: |a| = √(ax² + ay²)
Azimuth: φ = atan2(ay, ax)
Unit vector: â = a / |a|
Pythagorean theorem: |a|² = ax² + ay² + az²
$$|\mathbf{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}, \quad \hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$$How to Use
- Enter the three components of vector a (ax, ay, az).
- Set az to 0 for a 2D vector.
- The tool computes the magnitude and azimuth.
Case Studies
Force Magnitude and Speed
Resultant force: F=(30,40,0) N → |F|=50 N. Common in rope tension or inclined-plane force analysis; the 3-4-5 ratio is typical.
Speed: v=(20,15,0) m/s → |v|=25 m/s=90 km/h. Direction atan(15/20)=36.9°. A car speedometer shows speed (magnitude of velocity).
MTR (Hong Kong metro): straight segment v=(20,0), turning v=(15,15) → speed 21.2 m/s. Direction changes in curves while speed can stay constant.
Electric Field Strength and Navigation
Electric field: E=(3,4,0)×10⁴ N/C → |E|=5×10⁴ N/C. After superposition of multiple charges, compute total strength.
GPS navigation: position vector (1000,2000,50) m → |r|=2236 m. Horizontal distance √(1000²+2000²)=2236 m, height 50 m.
3D modeling: vector magnitude computes edge lengths and normals. Used in CAD, game engines and robot path planning.
FAQ
Why does magnitude use the Pythagorean theorem?
Vector components are mutually perpendicular (orthogonal), so |a|²=ax²+ay²+az² follows the Pythagorean theorem. The distance from the origin to point (ax,ay,az) is the vector magnitude. In 2D it is the planar distance, in 3D the spatial distance. Orthogonal components are independent and add in quadrature.
What is the difference between azimuth and elevation?
Azimuth: the angle in the xy-plane measured counterclockwise from the x-axis = atan2(ay,ax). Elevation: the angle with the xy-plane = arcsin(az/|a|). Azimuth describes the horizontal direction, elevation the vertical direction. Together they fully describe a 3D direction.
Can vector magnitude be negative?
No. |a|=√(sum of squares) is always non-negative. |0|=0. Magnitude is a length, physically a 'quantity'. For a signed direction use components. Speed = magnitude of velocity is always positive; velocity components can be positive or negative.
What is a unit vector?
A vector of magnitude 1: â=a/|a|. It keeps only direction, removing magnitude. Used to express direction. Coordinate unit vectors i=(1,0,0), j=(0,1,0), k=(0,0,1). Any vector a=|a|·â (magnitude times direction unit vector).
What are everyday uses of vector magnitude?
(1) Distance: magnitude of the displacement vector = straight-line distance. (2) Speed: magnitude of the velocity vector. (3) Magnitude of resultant force. (4) Electric/magnetic field strength. (5) Signal strength (Wi-Fi, phone signal dBm relates to field strength). Hong Kong GPS navigation and sports watches both compute vector magnitude.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.