Cross Product

Definition

The cross product of two vectors, denoted as $$a \times b$$, produces a third vector that is perpendicular to the plane formed by the original vectors. This relationship is expressed mathematically by the formula $$a \times b = |a||b|\sin\theta n$$, where $$|a|$$ and $$|b|$$ are the magnitudes of the vectors, $$\theta$$ is the angle between them, and $$n$$ is a unit vector perpendicular to both. The cross product captures important geometric properties of vectors and plays a critical role in physics, particularly in determining torque and rotational motion.

5 Must Know Facts For Your Next Test

1. The result of a cross product is always a vector that is orthogonal (perpendicular) to both original vectors.
2. The magnitude of the cross product $$|a \times b|$$ can be interpreted as the area of the parallelogram formed by vectors $$a$$ and $$b$$.
3. The direction of the resulting vector from a cross product follows the right-hand rule, which means if you curl your fingers from vector $$a$$ to vector $$b$$, your thumb points in the direction of $$n$$.
4. If the angle $$\theta$$ between vectors $$a$$ and $$b$$ is 0° or 180°, then their cross product is zero because $$\sin(0) = 0$$ and $$\sin(180) = 0$$.
5. The cross product is not commutative; that is, $$a \times b \neq b \times a$$. Instead, it is anti-commutative since $$a \times b = - (b \times a)$$.

Review Questions

• How does the geometric interpretation of the cross product aid in understanding its applications in physics?
  • The geometric interpretation of the cross product provides insight into its physical applications by relating it to areas and rotational forces. The magnitude of the cross product represents the area of the parallelogram formed by two vectors, which can be useful for calculating quantities like torque. Additionally, since the result is perpendicular to both original vectors, this helps in determining directions for forces and moments in mechanical systems.
• Discuss how the right-hand rule helps determine the direction of the resultant vector in a cross product operation.
  • The right-hand rule is a simple mnemonic used to find the direction of the resultant vector from a cross product. By orienting your right hand so that your fingers point along vector $$a$$ and then curling them toward vector $$b$$, your thumb will indicate the direction of the resultant vector $$n$$. This method not only helps visualize three-dimensional relationships between vectors but also reinforces why certain physical laws, such as those governing angular momentum or torque, rely on this specific orientation.
• Evaluate how changing the angle between two vectors affects their cross product and provide an example illustrating this principle.
  • Changing the angle between two vectors directly impacts their cross product through the sine function in the equation $$a \times b = |a||b|\sin\theta n$$. For example, if we have two vectors with equal magnitudes and one is rotated from 90° (where $$\sin(90) = 1$$) to 0° (where $$\sin(0) = 0$$), we observe that at 90°, their cross product yields maximum magnitude and points perpendicularly. However, at 0°, they become collinear, resulting in a zero cross product. This illustrates how angular relationships fundamentally affect rotational dynamics in various physical systems.

Related terms

Magnitude: The length or size of a vector, calculated as the square root of the sum of the squares of its components.

Unit Vector: A vector that has a magnitude of one and indicates direction, often used to specify the direction of a vector in space.

Torque: A measure of the rotational force acting on an object, which is calculated as the cross product of the position vector and the force vector.