Vector Product: Cross Product

Vector Product: Cross Product

Definition

• The cross product, also known as the vector product, is a binary operation that takes two vectors as input and produces another vector as output.
• It is denoted by the symbol × (cross) and is read as "a cross b" or "a times b".

Geometric Interpretation

• The cross product of two vectors a and b is a vector that is perpendicular to both a and b.
• The direction of the cross product is determined by the right-hand rule.
• The magnitude of the cross product is given by the area of the parallelogram spanned by the two input vectors.

Mathematical Formula

• The cross product of two vectors a = (a1, a2, a3) and b = (b1, b2, b3) is given by:

a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

Properties

• The cross product is anti-commutative, i.e., a × b = -b × a.
• The cross product is distributive over addition, i.e., a × (b + c) = a × b + a × c.

Applications

• The cross product is used to calculate the area of a parallelogram and the volume of a parallelepiped.
• It is also used in physics to calculate the torque and angular momentum of an object.

Vector Product: Cross Product

Definition

• The cross product, also known as the vector product, is a binary operation that takes two vectors as input and produces another vector as output.
• It is denoted by the symbol × (cross) and is read as "a cross b" or "a times b".

Geometric Interpretation

• The cross product of two vectors a and b is a vector that is perpendicular to both a and b.
• The direction of the cross product is determined by the right-hand rule.
• The magnitude of the cross product is given by the area of the parallelogram spanned by the two input vectors.

Mathematical Formula

• The cross product of two vectors a = (a1, a2, a3) and b = (b1, b2, b3) is given by: a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

Properties

• The cross product is anti-commutative, i.e., a × b = -b × a.
• The cross product is distributive over addition, i.e., a × (b + c) = a × b + a × c.

Applications

• The cross product is used to calculate the area of a parallelogram and the volume of a parallelepiped.
• It is also used in physics to calculate the torque and angular momentum of an object.

Learn about the definition and geometric interpretation of the vector product, also known as the cross product, a binary operation that takes two vectors as input and produces another vector as output.