A × b

Definition

The cross product, denoted as a × b, is a binary operation in vector algebra that produces a vector that is perpendicular to both of the input vectors, a and b. The cross product is an important concept in mathematics, physics, and various engineering applications, as it allows for the calculation of quantities such as torque, angular momentum, and the area of a parallelogram formed by the input vectors.

5 Must Know Facts For Your Next Test

1. The cross product of two vectors a and b is a vector that is perpendicular to both a and b, and its magnitude is equal to the area of the parallelogram formed by the input vectors.
2. The direction of the cross product is determined by the right-hand rule, where the thumb points in the direction of the resulting vector when the fingers of the right hand curl from the first vector to the second vector.
3. The cross product is anticommutative, meaning that a × b = -b × a.
4. The cross product is distributive over addition, so a × (b + c) = (a × b) + (a × c).
5. The cross product is useful in calculating quantities such as torque, angular momentum, and the normal vector to a surface.

Review Questions

• Explain the geometric interpretation of the cross product a × b.
  • The cross product a × b represents a vector that is perpendicular to both the input vectors a and b. The magnitude of the cross product is equal to the area of the parallelogram formed by the input vectors, and the direction of the cross product is determined by the right-hand rule. Specifically, if you point your right thumb in the direction of the first vector a, and curl your fingers from a to b, your outstretched right hand will indicate the direction of the cross product a × b.
• Describe the properties of the cross product, including its anticommutativity and distributivity.
  • The cross product has several important properties. First, it is anticommutative, meaning that a × b = -b × a. This reflects the fact that the direction of the cross product changes when the order of the input vectors is reversed. Additionally, the cross product is distributive over addition, so a × (b + c) = (a × b) + (a × c). This property allows for the cross product to be calculated efficiently using vector components. These properties of anticommutativity and distributivity are crucial in the application of the cross product to various mathematical and physical problems.
• Explain how the cross product is used to calculate important quantities in physics and engineering, such as torque, angular momentum, and the normal vector to a surface.
  • The cross product is a fundamental tool in physics and engineering for calculating various quantities. For example, the cross product of a position vector and a force vector gives the torque, which is the rotational analog of force and is crucial in the analysis of rotational motion. Similarly, the cross product of a position vector and a linear momentum vector gives the angular momentum, which is conserved in the absence of external torques. Additionally, the cross product of two vectors can be used to determine the normal vector to a surface, which is important in the study of surfaces and their properties. The versatility of the cross product makes it an indispensable concept in many areas of science and engineering.

Related terms

Vector:

A mathematical object that has both magnitude and direction, typically represented as an arrow in two or three-dimensional space.

Scalar:

A mathematical quantity that has magnitude but no direction, such as a real number.

Orthogonal: Two vectors are orthogonal if they are perpendicular to each other, meaning their dot product is zero.