2.8 Unit Vectors

Definition

A unit vector is a vector whose magnitude is exactly 1.

A unit vector is used to specify direction. It does not represent the magnitude of a vector.

Unit vectors provide a convenient way to express vectors in component form.

Standard Unit Vectors

In a Cartesian coordinate system, three standard unit vectors are defined along the positive coordinate axes.

Figure 2.8.1 – Standard unit vectors along the positive x-, y-, and z-axes.

î points in the positive x-direction.

ĵ points in the positive y-direction.

k̂ points in the positive z-direction.

Each of these vectors has a magnitude of 1.

|i^|=1|\hat{i}| = 1

|j^|=1|\hat{j}| = 1

|k^|=1|\hat{k}| = 1

Expressing Vectors with Unit Vectors

Any vector can be written as the sum of its components multiplied by the corresponding unit vectors. For a vector in two dimensions,

A→=Axi^+Ayj^\vec{A} = A_x\hat{i} + A_y\hat{j}

For a vector in three dimensions,

A→=Axi^+Ayj^+Azk^\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}

where

  • Aₓ is the x-component.
  • Ay is the y-component.
  • Az is the z-component.

This notation clearly shows both the magnitude of each component and its direction.

Example

Consider a vector whose components are

Aₓ = 4

Ay = 3

The vector can be written as

A→=4i^+3j^\vec{A} = 4\hat{i} + 3\hat{j}

Figure 2.8.2 – A vector expressed in terms of its horizontal and vertical unit vectors.

Its magnitude is

|A→|=42+32=5|\vec{A}| = \sqrt{4^2 + 3^2} = 5

Key Points

  • A unit vector has a magnitude of 1.
  • Unit vectors describe direction only.
  • î, ĵ, and k̂ point along the positive x-, y-, and z-axes.
  • Any vector can be expressed using its components and the corresponding unit vectors.

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2.9 Vector Addition →