The Torque of a ForceThe mechanical advantage that a familiar tool called a wrench provides ( Figure 2.31) depends on magnitude F of the applied force, on its direction with respect to the wrench handle, and on how far from the nut this force is applied. The anticommutative property means the vector product reverses the sign when the order of multiplication is reversed:
This means that vectors A → × B → A → × B → and B → × A → B → × A → are antiparallel to each other and that vector multiplication is not commutative but anticommutative. If we reverse the order of multiplication, so that now B → B → comes first in the product, then vector B → × A → B → × A → must point downward, as seen in Figure 2.29(b). In the standard right-handed orientation, where the angle between vectors is measured counterclockwise from the first vector, vector A → × B → A → × B → points upward, as seen in Figure 2.29(a). On the line perpendicular to the plane that contains vectors A → A → and B → B → there are two alternative directions-either up or down, as shown in Figure 2.29-and the direction of the vector product may be either one of them. (b) The vector product B → × A → B → × A → is a vector antiparallel to vector A → × B → A → × B →. Small squares drawn in perspective mark right angles between A → A → and C → C →, and between B → B → and C → C → so that if A → A → and B → B → lie on the floor, vector C → C → points vertically upward to the ceiling. (a) The vector product A → × B → A → × B → is a vector perpendicular to the plane that contains vectors A → A → and B → B →. Scalar multiplication of two vectors yields a scalar product.įigure 2.29 The vector product of two vectors is drawn in three-dimensional space. The Scalar Product of Two Vectors (the Dot Product) It is important to distinguish between these two kinds of vector multiplications because the scalar product is a scalar quantity and a vector product is a vector quantity. For example, in describing rotations, a vector quantity called torque is defined as a vector product of an applied force (a vector) and its distance from pivot to force (a vector).
Vector products are used to define other derived vector quantities. Taking a vector product of two vectors returns as a result a vector, as its name suggests.
A quite different kind of multiplication is a vector multiplication of vectors. For example, the work that a force (a vector) performs on an object while causing its displacement (a vector) is defined as a scalar product of the force vector with the displacement vector. Scalar products are used to define work and energy relations. Taking a scalar product of two vectors results in a number (a scalar), as its name indicates. One kind of multiplication is a scalar multiplication of two vectors. There are two kinds of products of vectors used broadly in physics and engineering.