0.1 Cartesian vectors and tensors: their algebra  (Page 3/5)

Since the vector product forms a right handed system, the product b × a has the same magnitude but opposite direction as a × b , i.e., the vector product is not commutative,

The vector product of a vector with itself or with a parallel vector is zero or the null vector, i.e., a × a = 0 . A quantity that is the negative of itself is zero. Also, the angle between parallel vectors is zero and thus the sine is zero.

Consider the vector product of the unit vectors. They are all of unit length and mutually orthogonal so their vector products will be unit vectors. Remembering the right-handed rule, we therefore have

The components of the vector product can be expressed in terms of the components of a and b and applying the above relations between the unit vectors.

The permutations of the indices and signs in the expression for the vector product may be difficult to remember. Notice that the expression is the same as that for the expansion of a determinate of the matrix,

Expansion of determinants are aided by the permutation symbol, ${\epsilon }_{ijk}$ .

The expression for the vector product is now as follows.

Velocity due to rigid body rotations

We will show that the velocity field of a rigid body can be described by two vectors, a translation velocity, v _(t) , and an angular velocity, ω . A rigid body has the constraint that the distance between two points in the body does not change with time. The translation velocity is the velocity of a fixed point, O , in the body, e.g., the center of mass. Now consider a new reference frame (coordinate system) with the origin at point O that is translating with respect to the original reference frame with the velocity v _(t) . The rotation of the body about O is defined by the angular velocity, ω , i.e., with a magnitude ω and a direction of the axis of rotation, n , such that the positive direction is the direction that a right handed screw advances when subject to the rotation, . ω =ω n . Consider a point P not on the axis of rotation, having coordinates x in the new reference frame. The velocity of P in the new reference frame has a magnitude equal to the product of ω and the radius of the point P from the axis of rotation. This radius is equal to the magnitude of x and the sine of the angle between x and ω , i.e., | x | sinθ. The velocity of point P in the new reference frame can be expressed as

The velocity field of any point of the rigid body in the original reference frame is now

Since this equation is valid for any pair of points in the rigid body, the relative velocity Δ v between a pair of points separated by Δ x can be expressed as follows.

Conversely, if the relative velocity between any pair of points is described by the above equation with the same value of angular velocity, then the motion is due to a rigid body rotation.