Vectors, Vector Product of Two Vectors, Vectors (Cross) Product of the Vectors

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Vector Product of Two Vectors:

Before we define vector product of two vectors, we discuss below right handed and left handed screw and associate it with corresponding vector triad.

Right Handed Screw:

If a screw is taken and rotated in the anticlockwise direction, it translates towards the reader. It is called right handed screw.

Left handed Screw:

  • If a screw is taken and rotated in the clockwise direction, it translates away from the reader. It is called a left handed screw.

  • Now we associate a screw with given ordered vector triad.

  • Let and be three vectors whose initial point is.

Left handed screw

Left Handed Screw

Left handed screw

  • Now if a right handed screw at is rotated from towards through an angle, it will undergo a translation along [Fig. (i)]

  • Similarly if a left handed screw at is rotated from to through an angle, it will undergo a translation along [Fig. (ii)]. This time the direction of translation will be opposite to the first one.

  • Thus an ordered vector triad is said to be right handed or left handed according as the right handed screw translated along or opposite to when it is rotated through an angle less than .

Vectors (Cross) Product of the Vectors:

If and are two non-zero vectors then their cross product is denoted by and defined as

Cross Product of the Vectors

Cross Product of the Vectors

Cross Product of the Vectors

Where is the angle between and, and is a unit vector perpendicular to both and ; such that ,and form a right handed system (see figure) i.e. the right handed system rotated from to moves in the direction of .

  • is a vector and .

  • If either or then is not defined and in that case we consider.

  • If and are non-zero vectors. Then if and only if and are collinear or parallel vectors. i.e. .

  • In particular and because in the first situation and in case. Making the value of in both the cases.

  • and .

  • .

  • .

  • Angle between two vectors and is given as

.

  • If and represent the adjacent sides of a triangle then its area is given by.

  • If and represent the adjacent sides of a parallelogram, then its area is given by.

  • If and , then

Unit vector perpendicular to both and is .

Example:

Find a unit vector perpendicular to each of the vectors and .

Solution:

Unit vector perpendicular to both and

Developed by: