Commerce > Examples : Vector Product

1. What is the vector product in mathematics? |

Ans. The vector product, also known as the cross product, is a binary operation performed on two vectors in three-dimensional space. It results in a vector that is perpendicular to both of the original vectors and its magnitude is equal to the product of their magnitudes multiplied by the sine of the angle between them.

2. How is the vector product calculated? |

Ans. To calculate the vector product of two vectors, we use the right-hand rule. The resulting vector is perpendicular to the plane formed by the two original vectors, and its direction is determined by curling the fingers of your right hand from the first vector to the second vector. The magnitude of the vector product is given by the product of the magnitudes of the original vectors and the sine of the angle between them.

3. What are the properties of the vector product? |

Ans. The vector product has several important properties. It is anti-commutative, meaning that the order of the vectors changes the direction of the resulting vector. It is also distributive over vector addition, which means that the vector product of two vectors added to the vector product of two other vectors is equal to the vector product of the sum of the first two vectors and the sum of the last two vectors. Additionally, the vector product of a vector with itself is always zero.

4. What is the geometric interpretation of the vector product? |

Ans. Geometrically, the vector product represents the area of the parallelogram formed by the two original vectors. The direction of the resulting vector is perpendicular to this parallelogram. If the two vectors are parallel or anti-parallel, the vector product will be zero, indicating that the parallelogram has no area.

5. How is the vector product used in physics and engineering? |

Ans. The vector product finds extensive applications in physics and engineering. It is used to calculate torque, which is the rotational analog of force, and is crucial in rotational dynamics. The vector product is also utilized in electromagnetism, where it helps determine the direction of magnetic fields resulting from current-carrying wires. Additionally, the vector product plays a role in calculating the magnetic moment of a current loop and in cross-sectional area calculations in fluid dynamics.

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