Question

What is a unit vector?

Short answer

A unit vector has a magnitude of one and indicates direction.

Step-by-step solution

Understanding Vectors

A vector is a quantity that has both magnitude (length) and direction. It is often represented as an arrow in a coordinate space. The length of the vector, or its magnitude, is typically calculated using the Euclidean norm.

Defining Unit Vectors

A unit vector is a vector that has a magnitude of exactly one. Unit vectors are often used to indicate direction without specifying magnitude. They maintain the direction of the original vector but are rescaled to have a length of one.

Notation and Examples

Unit vectors are usually denoted with a hat, for example, \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) for the unit vectors in the direction of the x-axis, y-axis, and z-axis respectively in 3D space. An example of converting any vector \( \vec{v} \) to a unit vector \( \hat{v} \) is by using the formula \( \hat{v} = \frac{\vec{v}}{\|\vec{v}\|} \), where \( \|\vec{v}\| \) is the magnitude of \( \vec{v} \).

Key concepts

Vectors

Vectors are a fundamental concept in mathematics and physics. Imagine arrows that point in a specific direction and have a particular length. These arrows, or vectors, are defined by both a 'magnitude', which is their length, and a 'direction', which is where they point.

Vectors are essential because they help represent quantities that aren't just about size but also about where they're going. For example, velocity and force are both described using vectors. In a coordinate system, you can represent vectors using components, such as \((x, y)\) in 2D or \( (x, y, z) \) in 3D.

• Each component of a vector describes its influence in a direction along the axes.
• Vectors can be added, subtracted, and scaled through these components, allowing for flexible manipulation in various calculations.

Magnitude

The magnitude of a vector is a measure of its length, calculated as the square root of the sum of the squares of its components. If you have a vector \( \vec{v} = (x, y) \), its magnitude, often denoted as \(|\vec{v}|\) or \(orm{\vec{v}}\), is determined using the formula: \(| \vec{v} | = \sqrt{x^2 + y^2} \).

This magnitude tells you how long the vector is regardless of its direction. It is an important property you need to understand before converting it into a unit vector.

• Larger magnitudes indicate longer vectors, while smaller ones mean shorter vectors.
• Magnitude is always a non-negative number, as it represents distance.

Direction

Determining the direction of a vector is crucial because vectors aren't just about how much; they also indicate where. The direction is often visualized through the angle the vector makes with a reference axis, such as the x-axis in a coordinate plane.

When you have a vector \(\vec{v} = (x, y)\), its direction can be described by the angle \( \theta \) it forms with the x-axis, which is computed using the tangent function: \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

• The direction specifies the orientation of the vector in space.
• Even for vectors of different lengths, if they point in the same direction, they have the same direction angle.

Euclidean norm

The Euclidean norm is a fancy term for something we use quite often—the magnitude of a vector. It stems from the length of the vector, calculated based on its components, using the Pythagorean theorem.

For a vector \( \vec{v} = (x, y, z) \) in 3D, the Euclidean norm \(orm{\vec{v}}\) is calculated as \(orm{\vec{v}} = \sqrt{x^2 + y^2 + z^2} \). This is the most common way people talk about the length of a vector.

• The Euclidean norm is always positive, reflecting the length of the vector in the geometrical space.
• Understanding the Euclidean norm is the first step in finding a unit vector, as you divide the vector by its Euclidean norm to get a vector of length one.