2.7 Vector Product (Cross Product) of Two Vectors | Vectors | Physics 11

Introduction

The vector product, also known as the cross product, is a binary operation performed on two vectors in three-dimensional space. Unlike the Scalar And Vector Quantities→ which describes the scalar (dot) product yielding a scalar result, the cross product produces a new vector. This resultant vector is perpendicular to the plane containing the original two vectors, making the cross product an essential tool for describing rotational motion, torque, angular momentum, and magnetic forces.

Definition and Formula

The cross product of two vectors, $A$ and $B$ , is denoted as $A \times B$ . The result is a vector, $C$ , defined as:

$A \times B = ∣ A ∣∣ B ∣ sin (θ) \overset{n}{^}$

Where:

$∣ A ∣$ and $∣ B ∣$ are the magnitudes of vectors $A$ and $B$ .
$θ$ is the smaller angle between the two vectors ( $0^{\circ} \leq θ \leq 18 0^{\circ}$ ).
$\overset{n}{^}$ is a unit vector perpendicular to the plane formed by $A$ and $B$ . Its direction is determined by the right-hand rule.

Magnitude of the Cross Product

The magnitude of the resultant vector is given by:

$∣ A \times B ∣ = A B sin (θ)$

Geometrically, this magnitude represents the area of the parallelogram formed by the two vectors $A$ and $B$ when placed tail-to-tail.

Direction: The Right-Hand Rule

The direction of the resultant vector $C = A \times B$ is perpendicular to both $A$ and $B$ and is found using the right-hand rule:

Point the fingers of your right hand in the direction of the first vector ( $A$ ).
Curl your fingers towards the direction of the second vector ( $B$ ) through the smaller angle.
Your thumb will point in the direction of the resultant vector ( $C = A \times B$ ).

Characteristics and Properties

Property	Description	Mathematical Expression
Anti-Commutative	Reversing the order of the vectors negates the resultant vector.	$A \times B = - (B \times A)$
Parallel Vectors	The cross product of parallel or anti-parallel vectors is the zero vector ( $θ = 0^{\circ}$ or $18 0^{\circ}$ ).	$A \times B = 0$
Self-Cross Product	The cross product of any vector with itself is the zero vector.	$A \times A = 0$
Perpendicular Vectors	The magnitude is maximized when vectors are perpendicular ( $θ = 9 0^{\circ}$ ).	$∥ A \times B ∥ = A B$
Distributive	The cross product is distributive over vector addition.	$A \times (B + C) = (A \times B) + (A \times C)$

Cross Product of Unit Vectors

For the orthogonal unit vectors $\hat{i}, \hat{j}, \hat{k}$ :

$\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$
$\hat{i} \times \hat{j} = \hat{k}$
$\hat{j} \times \hat{k} = \hat{i}$
$\hat{k} \times \hat{i} = \hat{j}$
$\hat{j} \times \hat{i} = - \hat{k}$

Vector Product using Rectangular Components

For practical calculations, the cross product is most easily computed using the components of the vectors in a right-handed coordinate system. Refer to Rectangular Components Of A Vector→ for more on components.

Given:

$A = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$

$B = B_{x} \hat{i} + B_{y} \hat{j} + B_{z} \hat{k}$

The cross product can be calculated using the determinant of a $3 \times 3$ matrix:

$A \times B = \hat{i} A_{x} B_{x} \hat{j} A_{y} B_{y} \hat{k} A_{z} B_{z}$

Expanding the determinant gives the resultant vector:

$A \times B = (A_{y} B_{z} - A_{z} B_{y}) \hat{i} - (A_{x} B_{z} - A_{z} B_{x}) \hat{j} + (A_{x} B_{y} - A_{y} B_{x}) \hat{k}$

Applications in Physics

Torque: Torque ( $τ$ ), the rotational equivalent of force, is defined as the cross product of the position vector ( $r$ ) and the applied force ( $F$ ). $τ = r \times F$
Angular Momentum: The angular momentum ( $L$ ) of a particle is the cross product of its position vector ( $r$ ) and its linear momentum ( $p$ ). $L = r \times p$
Magnetic Force: The force ( $F_{B}$ ) on a charge $q$ moving with velocity $v$ through a magnetic field $B$ is given by: $F_{B} = q (v \times B)$

Introduction

Definition and Formula

The cross product of two vectors, $A$ and $B$ , is denoted as $A \times B$ . The result is a vector, $C$ , defined as:

$A \times B = ∣ A ∣∣ B ∣ sin (θ) \overset{n}{^}$

Where:

$∣ A ∣$ and $∣ B ∣$ are the magnitudes of vectors $A$ and $B$ .
$θ$ is the smaller angle between the two vectors ( $0^{\circ} \leq θ \leq 18 0^{\circ}$ ).
$\overset{n}{^}$ is a unit vector perpendicular to the plane formed by $A$ and $B$ . Its direction is determined by the right-hand rule.

Magnitude of the Cross Product

The magnitude of the resultant vector is given by:

$∣ A \times B ∣ = A B sin (θ)$

Geometrically, this magnitude represents the area of the parallelogram formed by the two vectors $A$ and $B$ when placed tail-to-tail.

Direction: The Right-Hand Rule

The direction of the resultant vector $C = A \times B$ is perpendicular to both $A$ and $B$ and is found using the right-hand rule:

Point the fingers of your right hand in the direction of the first vector ( $A$ ).
Curl your fingers towards the direction of the second vector ( $B$ ) through the smaller angle.
Your thumb will point in the direction of the resultant vector ( $C = A \times B$ ).

Characteristics and Properties

Property	Description	Mathematical Expression
Anti-Commutative	Reversing the order of the vectors negates the resultant vector.	$A \times B = - (B \times A)$
Parallel Vectors	The cross product of parallel or anti-parallel vectors is the zero vector ( $θ = 0^{\circ}$ or $18 0^{\circ}$ ).	$A \times B = 0$
Self-Cross Product	The cross product of any vector with itself is the zero vector.	$A \times A = 0$
Perpendicular Vectors	The magnitude is maximized when vectors are perpendicular ( $θ = 9 0^{\circ}$ ).	$∥ A \times B ∥ = A B$
Distributive	The cross product is distributive over vector addition.	$A \times (B + C) = (A \times B) + (A \times C)$

Cross Product of Unit Vectors

For the orthogonal unit vectors $\hat{i}, \hat{j}, \hat{k}$ :

$\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$
$\hat{i} \times \hat{j} = \hat{k}$
$\hat{j} \times \hat{k} = \hat{i}$
$\hat{k} \times \hat{i} = \hat{j}$
$\hat{j} \times \hat{i} = - \hat{k}$

Vector Product using Rectangular Components

Given:

$A = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$

$B = B_{x} \hat{i} + B_{y} \hat{j} + B_{z} \hat{k}$

The cross product can be calculated using the determinant of a $3 \times 3$ matrix:

$A \times B = \hat{i} A_{x} B_{x} \hat{j} A_{y} B_{y} \hat{k} A_{z} B_{z}$

Expanding the determinant gives the resultant vector:

$A \times B = (A_{y} B_{z} - A_{z} B_{y}) \hat{i} - (A_{x} B_{z} - A_{z} B_{x}) \hat{j} + (A_{x} B_{y} - A_{y} B_{x}) \hat{k}$

Applications in Physics

Torque: Torque ( $τ$ ), the rotational equivalent of force, is defined as the cross product of the position vector ( $r$ ) and the applied force ( $F$ ). $τ = r \times F$
Angular Momentum: The angular momentum ( $L$ ) of a particle is the cross product of its position vector ( $r$ ) and its linear momentum ( $p$ ). $L = r \times p$
Magnetic Force: The force ( $F_{B}$ ) on a charge $q$ moving with velocity $v$ through a magnetic field $B$ is given by: $F_{B} = q (v \times B)$