1.4 Vectors  (Page 4/5)

Co-planar vectors

A pair of vectors determines an unique plane. The pair of vectors defining the plane and other vectors in that plane are called coplanar vectors.

Axial vector

Motion has two basic types : translational and rotational motions. The vector and scalar quantities, describing them are inherently different. Accordingly, there are two types of vectors to deal with quantities having direction. The system of vectors that we have referred so far is suitable for describing translational motion and such vectors are called “rectangular” or "polar" vectors.

A different type of vector called axial vector is used to describe rotational motion. Its graphical representation is same as that of rectangular vector, but its interpretation is different. What it means that the axial vector is represented by a straight line with an arrow head as in the case of polar vector; but the physical interpretation of axial vector differs. An axial vector, say $\mathbf{\omega }$ , is interpreted to act along the positive direction of the axis of rotation, while rotating anti –clockwise. A negative axial vector like, $-\mathbf{\omega }$ , is interpreted to act along the negative direction of axis of rotation, while rotating clockwise.

The figure above captures the concept of axial vector. It should be noted that the direction of the axial vector is essentially tied with the sense of rotation (clockwise or anti-clockwise). This linking of directions is stated with "Right hand (screw) rule". According to this rule ( see figure below ), if the stretched thumb of right hand points in the direction of axial vector, then the curl of the fist gives the direction of rotation. Its inverse is also true i.e if the curl of the right hand fist is placed in a manner to follow the direction of rotation, then the stretched thumb points in the direction of axial vector.

Axial vector is generally shown to be perpendicular to a plane. In such cases, we use a shortened symbol to represent axial or even other vectors, which are normal to the plane, by a "dot" or "cross" inscribed within a small circle. A "dot" inscribed within the circle indicates that the vector is pointing towards the viewer of the plane and a "cross" inscribed within the circle indicates that the vector is pointing away from the viewer of the plane.

Axial vector are also known as "pseudovectors". It is because axial vectors do not follow transformation of rectangular coordinate system. Vectors which follow coordinate transformation are called "true" or "polar" vectors. One important test to distinguish these two types of vector is that axial vector has a mirror image with negative sign unlike true vectors. Also, we shall learn about vector or cross product subsequently. This operation represent many important physical phenomena such as rotation and magnetic interaction etc. We should know that the vector resulting from cross product of true vectors is always axial i.e. pseudovectors vector like magnetic field, magnetic force, angular velocity, torque etc.