What do you mean by Vector Product?
The vector product, also known as the cross product, is an operation that combines two vectors to produce a new vector that is perpendicular to both of the original vectors. It is denoted by the symbol "×" or by the "cross" notation. The formula for the vector product of two vectors, A and B, in three-dimensional space is:
\(\vec{a} \times \vec{b} = |\vec{a}| \times |\vec{b}| \sin \mathrm{\theta} \, \hat{n}\)
In this formula:
- 'a' and 'b' are the two vectors being multiplied.
- |a| and |b| represent the magnitudes (lengths) of vectors 'a' and 'b', respectively.
- θ is the angle between vectors 'a' and 'b'.
- sin θ is the sine of the angle θ between the two vectors.
- \(\hat{n}\) is a unit vector perpendicular to both vectors, and its direction is determined by the right-hand rule.
The resulting vector, A × B, has a magnitude equal to |A| |B| sin(θ), and its direction is perpendicular to both A and B. The right-hand rule is used to determine the direction of the resulting vector. By convention, the order of the vectors in the cross product matters, meaning A × B and B × A can produce different results.
The vector product is used in various fields of mathematics and physics, such as calculating torque, finding normal vectors, and determining the direction of magnetic fields.
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