Vector Magnitude Calculator
Our free coordinate geometry calculator solves vector magnitude problems. Get worked examples, visual aids, and downloadable results.
Formula
|v| = sqrt(vx^2 + vy^2 + vz^2)
The magnitude (L2 norm) is the square root of the sum of the squares of all components. It represents the Euclidean length of the vector from the origin to the point defined by the components.
Worked Examples
Example 1: 3D Vector Magnitude
Problem: Find the magnitude of vector v = (3, 4, 12).
Solution: Magnitude = sqrt(3^2 + 4^2 + 12^2)\n= sqrt(9 + 16 + 144)\n= sqrt(169)\n= 13\nUnit vector = (3/13, 4/13, 12/13) = (0.2308, 0.3077, 0.9231)\nDirection angles: alpha = arccos(3/13) = 76.66 deg\n beta = arccos(4/13) = 72.08 deg\n gamma = arccos(12/13) = 22.62 deg
Result: |v| = 13 | Unit = (0.231, 0.308, 0.923) | Largest component: z (85.2%)
Example 2: Comparing Different Norms
Problem: Calculate L1, L2, and L-infinity norms for v = (6, -8, 0).
Solution: L1 (Manhattan): |6| + |-8| + |0| = 6 + 8 + 0 = 14\nL2 (Euclidean): sqrt(36 + 64 + 0) = sqrt(100) = 10\nL-inf (Chebyshev): max(|6|, |-8|, |0|) = max(6, 8, 0) = 8\nNote: L-inf <= L2 <= L1 always holds\nUnit vector = (0.6, -0.8, 0)\n2D angle = atan2(-8, 6) = -53.13 degrees
Result: L1 = 14 | L2 = 10 | L-inf = 8 | Angle = -53.13 deg
Frequently Asked Questions
What is vector magnitude and how is it calculated?
Vector magnitude, also called the norm or length of a vector, measures the total size or distance a vector represents. For a 2D vector (x, y), the magnitude is sqrt(x^2 + y^2). For a 3D vector (x, y, z), it is sqrt(x^2 + y^2 + z^2). This formula comes from the Pythagorean theorem extended to multiple dimensions. For example, the magnitude of (3, 4) is sqrt(9 + 16) = sqrt(25) = 5, and the magnitude of (3, 4, 12) is sqrt(9 + 16 + 144) = sqrt(169) = 13. The magnitude is always a non-negative real number and equals zero only for the zero vector. It is denoted |v| or ||v||.
What is the difference between magnitude and magnitude squared?
Magnitude squared is simply the sum of the squares of the components without taking the square root: |v|^2 = x^2 + y^2 + z^2. It equals the dot product of the vector with itself (v dot v). While magnitude gives the actual length, magnitude squared is often preferred in computational applications because it avoids the computationally expensive square root operation. When comparing distances or magnitudes, if you only need to know which is larger (not the actual values), comparing squared magnitudes gives the same answer. This optimization is widely used in game engines and physics simulations where millions of distance comparisons occur per frame.
What are the different types of vector norms?
The most common vector norms are the L1 norm (Manhattan or taxicab norm), L2 norm (Euclidean norm), and L-infinity norm (maximum or Chebyshev norm). The L1 norm sums the absolute values of all components: |x| + |y| + |z|. The L2 norm is the standard Euclidean magnitude: sqrt(x^2 + y^2 + z^2). The L-infinity norm takes the maximum absolute component: max(|x|, |y|, |z|). Each norm defines a different notion of distance and has different applications. L1 is used in compressed sensing and sparse signal recovery. L2 is used for standard geometric calculations. L-infinity is used in game AI for grid-based movement. In general, the Lp norm is (|x|^p + |y|^p + |z|^p)^(1/p).
How does magnitude relate to the unit vector?
A unit vector is obtained by dividing a vector by its magnitude, a process called normalization: u = v / |v|. The resulting unit vector has magnitude exactly 1 and points in the same direction as the original vector. Conversely, any vector can be expressed as its magnitude times its unit vector: v = |v| * u. This decomposition separates the size information (magnitude) from the directional information (unit vector). This is incredibly useful in physics where you might need to specify a direction independently of a quantity. For example, a force vector F = |F| * F-hat, where |F| is the force magnitude and F-hat is the direction.
What do direction angles and direction cosines tell you about a vector?
Direction angles are the angles a vector makes with each coordinate axis. For a vector in 3D, alpha is the angle with the x-axis, beta with the y-axis, and gamma with the z-axis. Direction cosines are the cosines of these angles: cos(alpha) = x/|v|, cos(beta) = y/|v|, cos(gamma) = z/|v|. These are exactly the components of the unit vector. An important identity is cos^2(alpha) + cos^2(beta) + cos^2(gamma) = 1. Direction cosines fully specify the orientation of a vector in space. They are used in structural engineering for force resolution, in crystallography for describing lattice directions, and in aerospace engineering for attitude determination.
How is vector magnitude used in physics?
In physics, vector magnitude extracts the scalar quantity from a vector quantity. The magnitude of a velocity vector gives speed. The magnitude of a displacement vector gives distance. The magnitude of a force vector gives force intensity. The magnitude of an acceleration vector gives the rate of speed change. Kinetic energy depends on the magnitude of velocity: KE = 0.5*m*|v|^2. The magnitude of the electric field determines force on a charge: F = q*|E|. Gravitational potential energy relates to the magnitude of the position vector from the center of the Earth. Understanding magnitude is essential for converting between vector and scalar descriptions of physical phenomena.