In vector algebra, the cross product, or vector product of two vectors, is a mathematical procedure. It is important to characterise concepts like angular momentum and torque. A new vector perpendicular to the plane formed by the initial two vectors is produced by the vector product. Only in three dimensions this process can be used.
Definition
The vector product of two vectors A and B is defined as:
A×B = ∣A∣ ∣B∣ sin θ n Here:
∣A∣ and ∣B∣ are the magnitudes of A and B, respectively.
θ is the angle between A and B (measured in the range
0 ≤ θ ≤ π).
Sin θ is the sin of the angle between the two vectors.
n is a unit vector perpendicular to the plane formed by A and B, following is the right-hand rule.
Right-Hand Rule
The right-hand rule is used to identify the direction of the resulting vector A × B:
1. Start by pointing your right hand’s fingers toward A.
2. Use the smaller angle to twist them in the direction of B.
3. Your thumb gestures toward A and B.
Properties of Vector Product
1.Magnitude: The magnitude of A × B is given by:
∣A × B∣ = ∣A∣ ∣B∣ sinθ
The maximum value is obtained when θ = 90∘, and the minimum value is zero when θ = 0∘ or θ =180∘.
2. Direction: The resultant vector is always perpendicular to the plane containing A and B.
3. Non-Commutativity: A × B = −(B × A)
The cross product is anti-commutative because reversing the order changes the direction of the resultant vector.
4. Distributive Property: A × (B+C) = A × B + A × C
5. Self-Cross Product: A × A = 0
This means that the cross product of a vector with itself is always zero
Vector Product in Component Form
If A = Axi + Ayj + Azk, and B = Bxi + Byj + Bzk, the vector product is given by:
A × B = i j k
Ax Ay Az
Bx By Bz
Expanding this determinant give way:
A × B = (Ay Bz − AzBy) I − (Ax Bz − Az Bx) j +(AxBy−AyBx)k
Applications of Vector Product
1.Torque: In mechanics, torque (τ) is defined as the cross product of the force vector F and the position vector r: T = r × F
2. Angular Momentum: The angular momentum (L) of a particle is given by: L = r × p where p is the linear momentum.
3. Electromagnetic Forces: The force on a charged particle moving in a magnetic field is determined using the vector product:
F = qv × B
where q is the charge, v is the velocity, and B is the magnetic field.
Note :-
A basic process in vector algebra, the vector product has numerous uses in engineering and science. In addition to provides a way to compute variables essential for rotational dynamics and electromagnetic phenomena, it also deals the interaction between vectors in three dimensions.
Its characteristics and uses can significantly improve one’s ability to solve problems in the scientific and engineering fields.
A procedure between two vectors that creates a third vector perpendicular to the plane created by the original vectors is called a vector product, sometimes called as a cross product. In mathematics, it is defined as:
A × B = ∣A∣ ∣B∣ sin θ n
where ∣A∣ and ∣B∣ are the magnitudes of the vectors, θ is the angle between them, and n is the unit vector perpendicular to the plane.
The resulting vector’s direction is perpendicular to the plane that the two original vectors formed. The right-hand rule is used to determine its precise orientation.
Point the fingers of your right hand in the direction of the first vector (A).
Twist them towards the second vector (B).
Your thumb points in the direction of A × B.
The sine function (sin θ) accounts for the angle between the two vectors and determines the magnitude of the cross product. When θ = 90∘, sin θ = 1, and the cross product is maximised. If θ = 0∘ or 180∘, sin θ = 0, and the cross product is zero, meaning the vectors are parallel or anti-parallel.
When two vectors are parallel or anti-parallel, the angle between them (θ) is 0∘ or 180∘. Since sin 0∘ = 0 and sin 180∘ = 0, the cross product becomes zero: A × B = 0
Key properties of the vector product are:
Anti-Commutative: A × B = −(B × A)
Distributive: A × (B + C) = A × B + A × C
Scalar Multiplication: (kA) × B = k (A × B), where k is a scalar.
Zero Cross Product: A × A = 0.
A = Axi + Ayj + Azk, the cross product is:
For vectors A = Axi + Ayj + Azk, and B = BxI + Byj + Bzk, the vector product is given by:
A × B = I j k
Ax Ay Az
Bx By Bz
Expanding this determinant gives the resultant vector in terms of I,j,k.
The vector product has various applications in physics and engineering:
Torque: τ = r × F, where is the position vector and F is the force.
Angular Momentum: L = r × p, where p is the linear momentum.
Electromagnetic Force: F = qv × B, where q is charge, v is velocity, and B is the magnetic field.
These applications help describe rotational and electromagnetic phenomena.