How to Find a Unit Vector: Definition, Equation & Examples (2024)

Co-authored byDev Murphy, MAReviewed byGrace Imson, MA

Last Updated: June 19, 2023Fact Checked

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1. The formula to find unit vector is . You can calculate the unit vector of any vector by dividing the vector by its magnitude . For instance, to calculate the unit vector of vector , you would follow this equation: .^[1]

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   Identify the magnitude of the vector. The magnitude of a vector is notated as . To identify the magnitude of a 2-dimensional vector, you must first identify the horizontal and vertical components of the vector—that is, identify the coordinates of the vector along the x-axis and y-axis (for instance, 3 on the x-axis and 4 on the y-axis, usually notated as and , respectively) and find the sum of the squares of each of the components (which would be expressed as ). Take that sum, and find its square root in order to determine the magnitude. In other words, = √ = √ = .^[2]

   • To determine the magnitude of a 3-dimensional vector, follow the exact same formula, except add the z-axis—that is, the axis that usually intersects with the x- and y-axes diagonally (notated as ).
   • For instance, if 3-dimensional vector = , you would determine using the following equation: = √.
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   See Also

   Unit Vector- Definition, Formula, Example and Solved ProblemMagnitude of a Vector - Formula, How to Find? | Length of VectorUnit Vector - Formula, Definition, Caculate, Notation

   Divide the vector by its magnitude. Once you’ve determined the magnitude of the vector, take the vector and divide it by its magnitude. So if vector is , then = = (, ). The unit vector, then, is ().^[3]

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1. A vector is an object that contains both direction and magnitude. A vector is represented as a line on the Cartesian coordinate system, and it is denoted as a set of numbers representing its relationship to the x-axis (the vertical line on the graph) and the y-axis (the horizontal line). A vector’s magnitude refers to the size of the vector and is represented by a line with an arrow on the end. The length of the line denotes the length of the vector, while the direction of the arrow denotes the direction of the vector. Vectors possess both a starting point (the tail) and an ending point (the head, where the arrow is).^[4]

   • The vector is identified by its intersecting axes, the x-axis (usually represented by the mathematical notation ) and the y-axis (notated as ). Thus, the vector may be formulated as = .
   • Vectors are used in various arithmetic operations including addition, subtraction, and multiplication.
   • Vectors are also sometimes called Euclidean vectors, geometric vectors, or spatial vectors.

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1. A unit vector is a vector with a magnitude of 1. Unit vectors, also called direction vectors, are used to describe the direction of a given vector—or, the angle the vector makes on an x-axis. Unit vectors are marked with a cap symbol, which looks like a little arrow pointing upward: ^.^[5]

   • For example, if , , and are the unit vectors in the directions of the x-axis, the y-axis, and the z-axis, they would be denoted as , , and , respectively.

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1. The magnitude is the length or size of the vector. A vector’s magnitude, also called its norm, denotes its length on an x-axis (or the vertical line on the graph on which the vector appears), giving it numeric value. The magnitude on summarizes the length of the vector on the x-axis, y-axis (the horizontal line), and z-axis. Magnitude is denoted as 2 vertical lines, one on each side of the vector: . To calculate the magnitude of a given vector with the direction of the x-axis, y-axis, or z-axis, follow the Pythagorean Theorem: add the squares of the vector’s direction ratios. Then, calculate the square root of that sum.^[6]

   • The magnitude of would be , the magnitude of would be , and the magnitude of would be .
   • To find the magnitude of vector , where = (and where , , and represent the x-axis, y-axis, and z-axis, respectively), use the following equation: = √.
   • For example, if and and , then , and = √ = √ = . Thus, the magnitude of vector would be 3 units.

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   Given vector = + + , find .

2. 2

   Given vector = + , find .

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   Answer #1: =. In this problem, we're asked to find the unit vector of a 3-dimensional vector; hence, the inclusion of the z-axis in the equation. To find the unit vector, we identified the magnitude of the vector and then divided the vector by its magnitude:^[7]

   • = √ =
   • = =
   • =
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   Answer #2: = . For this problem, we're only asked to calculate and . Just as with the previous problem, start by calculating the magnitude using the Pythagorean Theorem. Then divide the vector by the magnitude:^[8]

   • = √ =
   • = =
   • =

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