6.5. VECTOR PRODUCT OF TWO VECTORS

We are already familiar with vectors and their use in physics. In chapter 5 (Work, Energy, Power) we defined the scalar product of two vectors. An important physical quantity, work, is defined as a scalar product of two vector quantities, force and displacement.

In this section, we introduce the concept of the vector product, which is different from the scalar product. While the scalar product results in a single quantity (a scalar), the vector product results in another vector. This distinction is crucial for understanding how different physical quantities interact, particularly in rotational motion.

Think of the scalar product as finding out how much work you did when pushing a box across the floor—it’s a single measure of energy. In contrast, the vector product is like determining the direction of the torque when you twist a doorknob: it not only matters how much force you apply but also the direction in which you apply that force.

A vector product of two vectors a and b is a vector c such that (i) magnitude of c = c sin(abθ), where a and b are magnitudes of a and b and θ is the angle between the two vectors. (ii) c is perpendicular to the plane containing a and b. (iii) if we take a right handed screw with its head lying in the plane of a and b and the screw perpendicular to this plane, and if we turn the head in the direction from a to b, then the tip of the screw advances in the direction of c. This right handed screw rule is illustrated in Fig. 6.15a.

The vector product (also called the cross product) defines a new vector that is perpendicular to both vectors involved in the operation. The magnitude of this vector product is determined by the sine of the angle between the two original vectors, emphasizing the role of direction in vector operations. This property is essential for calculating torque and angular momentum.

Imagine using a screwdriver: the direction in which you turn the screwdriver (whether clockwise or counterclockwise) affects how it drives a screw into the wood. The right-hand screw rule illustrates this physical behavior where the direction of the resulting torque vector depends on the direction you turn the handle.

Note that scalar product of two vectors is commutative as said earlier, a.b = b.a. The vector product, however, is not commutative, i.e., a × b ≠ b × a. The magnitude of both a × b and b × a is the same (sin(abθ)); also, both of them are perpendicular to the plane of a and b. But the rotation of the right-handed screw in the case of a × b is from a to b, whereas in the case of b × a it is from b to a. This means the two vectors are in opposite directions.

The commutative property of the scalar product allows you to swap the order of the vectors without changing the result. However, for vector products, changing the order flips the direction of the resulting vector. This property is essential for understanding operations involving rotations and actions in physics. Furthermore, this section introduces the impact of reflection on vector products, which helps to understand how vectors behave under various transformations.

This difference can be likened to two people pushing a cart in opposite directions: if they were to swap places, the direction of the movement would shift dramatically, resulting in a different outcome entirely. Understanding this helps when analyzing situations where forces and rotational effects are at play.

We may write c = a × b in the component form. For this we first need to obtain some elementary cross products: (i) a × a = 0 (0 is a null vector, i.e. a vector with zero magnitude). This follows since the magnitude of a × a is 2sin0 0a° =. From this follow the results (i) i × i = 0, j × j = 0, k × k = 0. (ii) i × j = k. Note that the magnitude of i × j is sin90° or 1, since i and j both have unit magnitude and the angle between them is 90°.

This chunk discusses how vector products can be expressed through their components. Understanding the specific relationships between unit vectors forms the basis for further calculations in physics. Knowing that the product of a vector with itself equals zero reflects an essential property of vectors that simplifies many calculations. Additionally, understanding the unit vectors and their relations aids in visualizing and solving 3D problems effectively.

Think of the unit vectors as the basic colors of light; just as combining different colors gives rise to new ones, the interaction between these fundamental vectors forms the basis for constructing complex vector descriptions in physics.

An interesting property of a vector product is its behavior under reflection. Under reflection (i.e. on taking the plane mirror image) we have, and, x x y y z z→ − → − → −. As a result, all the components of a vector change sign and thus, a → − b b → −. What happens to a × b under reflection? a × b → − × − = a × b. Thus, a × b does not change sign under reflection.

The reflection property of vector products shows how these vectors respond to changes in their orientation. This behavior is crucial in physics, particularly in understanding forces and torque in various systems. By recognizing this property, students can better grasp how vectors interact with their surroundings, which connects to symmetry and conservation laws.

Visualize looking in a mirror: as you raise one hand, your reflection mimics this action. Just as the vectors behave predictably under reflection, understanding these relationships aids in comprehending the fundamental principles governing the physical world.

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