Unit Vector Calculator
Free Unit vector Calculator for coordinate geometry. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
u = v / |v| where |v| = sqrt(vx^2 + vy^2 + vz^2)
The unit vector u is found by dividing each component of vector v by its magnitude |v|. The resulting vector has magnitude 1 and points in the same direction as the original.
Worked Examples
Example 1: Unit Vector in 2D
Problem:Find the unit vector for v = (3, 4).
Solution:Magnitude |v| = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5\nUnit vector u = v / |v| = (3/5, 4/5) = (0.6, 0.8)\nVerification: |u| = sqrt(0.6^2 + 0.8^2) = sqrt(0.36 + 0.64) = sqrt(1.00) = 1\nDirection angle from x-axis = atan2(4, 3) = 53.13 degrees
Result:Unit vector = (0.6, 0.8) | Magnitude = 5 | Angle = 53.13 deg
Example 2: Unit Vector in 3D
Problem:Find the unit vector for v = (2, -3, 6).
Solution:Magnitude |v| = sqrt(2^2 + (-3)^2 + 6^2) = sqrt(4 + 9 + 36) = sqrt(49) = 7\nUnit vector u = (2/7, -3/7, 6/7) = (0.2857, -0.4286, 0.8571)\nDirection angles: alpha = arccos(2/7) = 73.40 deg\n beta = arccos(-3/7) = 115.38 deg\n gamma = arccos(6/7) = 31.00 deg\nVerification: cos^2(alpha) + cos^2(beta) + cos^2(gamma) = 1
Result:Unit vector = (0.2857, -0.4286, 0.8571) | Magnitude = 7
Frequently Asked Questions
What is a unit vector and why is it important?
A unit vector is a vector with a magnitude (length) of exactly 1 that points in the same direction as the original vector. It is found by dividing each component of the vector by the vectors magnitude. Unit vectors are fundamental in mathematics and physics because they represent pure direction without any magnitude information. They are used to define coordinate systems (the standard basis vectors i, j, k are unit vectors), specify directions for forces and velocities, create normal vectors for surfaces, and define orientation in computer graphics. Any nonzero vector can be converted to a unit vector through normalization.
How do you calculate a unit vector from a given vector?
To calculate a unit vector, first find the magnitude of the original vector using the formula |v| = sqrt(vx^2 + vy^2 + vz^2). Then divide each component by this magnitude: the unit vector u = (vx/|v|, vy/|v|, vz/|v|). For example, for vector (3, 4), the magnitude is sqrt(9 + 16) = sqrt(25) = 5, so the unit vector is (3/5, 4/5) = (0.6, 0.8). You can verify correctness by checking that the resulting vector has magnitude 1: sqrt(0.6^2 + 0.8^2) = sqrt(0.36 + 0.64) = sqrt(1) = 1. This process is called normalization, and the resulting vector is called the normalized vector.
What are the standard basis unit vectors i, j, and k?
The standard basis unit vectors are i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1) in three-dimensional space. Each points along one of the coordinate axes with magnitude 1. Any vector in 3D space can be expressed as a linear combination of these basis vectors: v = vx*i + vy*j + vz*k. For example, the vector (3, -2, 5) equals 3i - 2j + 5k. These basis vectors are mutually orthogonal (perpendicular to each other) and their dot products follow i dot i = 1, i dot j = 0, and their cross products follow i cross j = k (right-hand rule). They form the foundation of the Cartesian coordinate system used throughout science and engineering.
How are unit vectors used in physics?
Unit vectors are used extensively in physics to separate the concepts of magnitude and direction. In mechanics, forces are often decomposed into components using unit vectors: F = F_x*i + F_y*j + F_z*k. In electromagnetics, the direction of electric and magnetic fields is expressed using unit vectors, and Coulombs law uses the unit vector r-hat to specify the direction from one charge to another. In orbital mechanics, the radial and tangential unit vectors describe satellite motion. Unit vectors are essential for defining coordinate systems in polar, cylindrical, and spherical coordinates, where the direction of the unit vectors changes with position, unlike the constant Cartesian basis vectors.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy