A × b

Definition

The expression 'a × b' represents the vector cross product of two vectors, denoted as 'a' and 'b'. This operation results in a new vector that is perpendicular to both of the original vectors and follows the right-hand rule for its direction. The magnitude of the resulting vector is determined by the area of the parallelogram formed by vectors 'a' and 'b', emphasizing its geometric significance in physics and engineering.

5 Must Know Facts For Your Next Test

1. The cross product is defined only in three-dimensional space, unlike the dot product which can be applied in any dimension.
2. The result of the cross product, 'a × b', is a vector that has a direction given by the right-hand rule: if you point your fingers of your right hand in the direction of 'a' and curl them towards 'b', your thumb points in the direction of 'a × b'.
3. The magnitude of 'a × b' can be calculated using the formula $$|a imes b| = |a||b| \, sin(\theta)$$, where $$\theta$$ is the angle between the two vectors.
4. The cross product is anti-commutative, meaning that $$a imes b = - (b imes a)$$.
5. The cross product is distributive over vector addition, so for any vectors 'a', 'b', and 'c', we have $$a \times (b + c) = (a \times b) + (a \times c)$$.

Review Questions

• How does the direction of the resulting vector from the cross product relate to the original vectors?
  • The direction of the vector resulting from the cross product 'a × b' is perpendicular to both original vectors 'a' and 'b'. This perpendicularity can be visualized using the right-hand rule: by orienting your right hand with your fingers pointing in the direction of vector 'a' and curling them toward vector 'b', your thumb will indicate the direction of 'a × b'. This relationship is crucial in applications such as torque and angular momentum in physics.
• In what scenarios would you prefer using the cross product over the dot product when analyzing two vectors?
  • The cross product is preferred when you need to find a vector that is orthogonal to two given vectors, such as when calculating torque or determining angular momentum. In contrast, the dot product provides a scalar value indicating how aligned two vectors are, which is useful for work calculations or projections. Thus, choosing between these operations depends on whether you need a new vector or a scalar representation of interaction between two vectors.
• Evaluate how the properties of the cross product can be applied in solving real-world problems involving rotational motion.
  • The properties of the cross product are essential in understanding and solving problems related to rotational motion. For instance, torque is defined as $$\tau = r \times F$$, where 'r' is the position vector from pivot to point of force application, and 'F' is the force vector. The resulting torque vector indicates not only how much force is applied but also its effectiveness at causing rotation about a pivot point. By analyzing forces using cross products, engineers can design systems with optimal rotational dynamics, like gears and levers.

Related terms

Vector: A quantity that has both magnitude and direction, represented graphically by an arrow.

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

Magnitude: The length or size of a vector, representing how much of that quantity there is.