Cross product

Definition

The cross product is a mathematical operation on two vectors in three-dimensional space that produces a third vector which is orthogonal to both of the original vectors. This operation not only helps to find the direction and area of parallelograms defined by the two vectors, but it also plays a significant role in calculating torque and angular momentum in physics.

5 Must Know Facts For Your Next Test

1. The cross product of two vectors $ extbf{a}$ and $ extbf{b}$ is given by the formula: $ extbf{a} \times \textbf{b} = ||\textbf{a}|| ||\textbf{b}|| \sin(\theta) \hat{n}$, where $\theta$ is the angle between them and $\hat{n}$ is the unit vector perpendicular to the plane formed by $ extbf{a}$ and $ extbf{b}$.
2. The resulting vector from a cross product has a magnitude equal to the area of the parallelogram formed by the two original vectors.
3. The direction of the cross product vector can be determined using the right-hand rule: if you point your right thumb in the direction of $ extbf{a}$ and your fingers in the direction of $ extbf{b}$, your palm will face in the direction of $ extbf{a} \times \textbf{b}$.
4. The cross product is not commutative, meaning that $ extbf{a} \times \textbf{b}$ is not equal to $ extbf{b} \times \textbf{a}$; instead, $ extbf{a} \times \textbf{b} = - (\textbf{b} \times \textbf{a})$.
5. In three-dimensional space, the cross product can be used to find normal vectors for surfaces, which is essential for understanding surface area calculations and parametric surfaces.

Review Questions

• How does the cross product help to determine the orientation and area of a parallelogram formed by two vectors?
  • The cross product provides both the magnitude and direction of a vector that represents the area of a parallelogram defined by two vectors. The magnitude of this vector equals the area calculated as $||\textbf{a}|| ||\textbf{b}|| \sin(\theta)$, where $\theta$ is the angle between the vectors. Moreover, since the resulting vector from the cross product is orthogonal to both original vectors, it indicates the orientation of this parallelogram in three-dimensional space.
• Compare and contrast the properties of the cross product with those of the dot product, particularly in terms of directionality and application.
  • While both operations involve two vectors, they yield different types of results and serve distinct purposes. The dot product results in a scalar that reflects how much two vectors align, capturing directional similarity without producing a new vector. In contrast, the cross product generates a new vector that is orthogonal to both inputs, representing concepts like torque or angular momentum. This distinction makes each operation useful for specific applications: dot products for projections and angles, while cross products are essential for areas and rotational dynamics.
• Evaluate how understanding cross products influences calculations regarding surface areas in parametric surfaces and their normals.
  • Understanding cross products is crucial when calculating surface areas defined by parametric equations. The normal vector obtained through cross products allows us to determine how surfaces interact with light or forces acting upon them. For example, when calculating a surface's area parametrically, finding tangent vectors at each point enables us to use their cross product to derive an area element. This understanding helps solve problems involving flux through surfaces or optimization of surface shapes based on their orientations.

Related terms

Vector: A quantity that has both magnitude and direction, represented as an arrow in space.

Dot Product: An operation that takes two vectors and returns a scalar value, indicating the extent to which they point in the same direction.

Magnitude: The length or size of a vector, representing how far it extends in space.