The vector product is also called the cross product.
When two vectors are multiplied using the cross product, the result is a new vector.
This new vector is always perpendicular to the plane formed by the original two vectors.
The cross product is only defined for two vectors and always results in another vector.
Formula: A × B = AB sinθ n̂
A and B are the magnitudes of the two vectors.
θ is the angle between the two vectors.
n̂ is the unit vector that shows the direction of the resulting vector, which is perpendicular to both A and B.
Direction and Properties of Vector Product
The direction of the cross product is determined by the right-hand rule: if you point your fingers from A to B, your thumb points in the direction of A × B.
A × B is not equal to B × A; the direction is reversed.
The magnitude of A × B and B × A is the same, but their directions are opposite.
Anti-commutative property: A × B = – (B × A)
This means reversing the order of the vectors reverses the direction of the result.
Examples of Vector Product
Torque (τ):
Formula: τ = r × F
Here, r is the position vector and F is the force vector.
The result, torque, is a vector with a specific direction.
Angular Momentum:
Formula: L = r × p
r is the position vector and p is the momentum vector.
The result, angular momentum, is also a vector with a defined direction.
In both cases, the direction of the result is important and follows the right-hand rule.
Properties of Vector Product
Product of Parallel or Anti-parallel Vectors:
If two vectors are parallel (angle 0°) or anti-parallel (angle 180°), their cross product is zero.
This is because sin(0°) = 0 and sin(180°) = 0.
Product of a Vector with Itself:
A × A = 0 because the angle between the same vector is 0°.
Product of Perpendicular Vectors:
If two vectors are perpendicular (angle 90°), the cross product is maximum.
A × B = AB because sin(90°) = 1.
Summary Table:
Parallel or anti-parallel: cross product = 0
Perpendicular: cross product = AB
Same vector: cross product = 0
Cross Product with Unit Vectors
The cross product of standard unit vectors follows these rules:
i × j = k
j × k = i
k × i = j
The direction follows the right-hand rule.
If the order is reversed, the result is negative:
j × i = –k
k × j = –i
i × k = –j
Cross Product in Rectangular Components
The cross product can be calculated using the determinant method:
Write the components of vectors A and B in terms of i, j, and k.
Set up a 3x3 determinant with the first row as i, j, k; the second row as the components of A; and the third row as the components of B.
Expanding the determinant gives the cross product in component form.
Physical Significance
The magnitude of the cross product of two vectors gives the area of the parallelogram formed by those vectors.
|A × B| = Area of parallelogram
The direction of the resulting vector (given by the unit vector n̂) is perpendicular (normal) to the plane containing A and B.
In simple terms, the cross product not only gives a vector but also represents the area and orientation of the parallelogram formed by the two vectors.
Key Terms & Definitions
Vector Product (Cross Product): The product of two vectors resulting in a new vector perpendicular to both.
Anti-commutative Property: The cross product reverses direction if the order of vectors is switched: A × B = – (B × A).
Determinant Method: A way to calculate the cross product using the components of the vectors.
Torque: The rotational effect of a force, calculated as r × F.
Angular Momentum: The rotational momentum of a body, calculated as r × p.
Unit Vector (n̂): Indicates the direction of the cross product, perpendicular to the plane of the original vectors.
Action Items / Next Steps
Memorize the definition, formula, and properties of the vector (cross) product.
Practice examples, especially torque and angular momentum.
Learn the cross product rules for unit vectors (i, j, k).
Understand and practice the determinant method for finding the cross product in component form.
Watch the video link provided in the description for more MCQs and short questions.
Prepare for the next topic by reviewing MCQs and short questions related to the cross product.