Loading....
Coupon Accepted Successfully!

 

Vector Product of Vectors

Two vectors can be multiplied in such a way as to yield a vector. In this case, the product is called the vector (or cross-product).
Let the vector product of two vectors A and B be equal to a vector C, i. e.
A B = C
Which is read as 'A cross B is equal to C'. 
The vector product is represented by putting a cross () between the vectors. 
The magnitude C of the resulting vector C is defined by
C = AB sin θ
where A and B respectively are the magnitudes of the vectors A and B and θ is the angle between them.

The direction of C is perpendicular to the plane formed by A and B. 

If vectors A and B lie in the plane of this page, the vector C will be perpendicular to this plane. 

The sense (upward or downward) of the direction of the vector product is given by the direction of the advance of the tip of a right-handed screw when rotated from A to B through angle θ between them, the screw being placed with its axis perpendicular to the plane containing the two vectors. 

This is known as the right-handed screw rule.

Another simple method of determining the direction of the vector product is as follows.

Hold your right hand with the thumb erect and the fingers curled. The direction of the thumb is perpendicular to the plane containing vectors A and B. If the direction of the rotation of the vector from A to B is the same as the direction of the folding of the fingers, then the erect thumb points in the direction of A B.

We say that the vectors A, B and C in the definition C = A B form a right handed coordinate system.
The vector product of two vectors A and B is a single vector C whose magnitude I equal to the product of the magnitude of vectors A and B and the sine of the angle between them and whose direction is perpendicular to the plane containing A and B in such a way that A, B and C form a right handed coordinate system.

This is the geometric way of defining a vector product. The algebraic definition of a vector product gives the components of (A B) in terms of those of A and B:
(A B)x = AyBz - AzBy
(A B)y = AzBx - AxBz
(A B)z = AxBy - AyBx

For a right handed orthogonal coordinate system, the two definitions of A ⋅ B give the same result. Which definition we use in a particular problem depends upon convenience.

Properties of a Vector Product

  1. The vector product is anti commutative, i. e. A B = - B A. This can be proved as follows. Let
    C = A B
    and D = B A
    From the definition of the vector product, the magnitude of C and D are
    C = AB sin θ
    D = BA sin θ
    where A and B are the magnitudes of A and B, and θ is the angle between them. Since A and B are scalars, AB = BA. Hence, the magnitude of C is equal to that of D. But their directions are different.
    To find their directions we use the right hand screw rule.
    Imagine that A and B lie in the plane of the paper. Place a right handed screw perpendicular to this plane. Rotate it from A to B to get the direction of A B.
    The tip will advance upward (i. e. towards the reader). This is the direction of C = A B.
    Now rotate it from B to A (to get the direction of B A); the tip will advance downwards (i. e. into the page). This is the direction of D = B A. Thus, the directions of C and D are opposite; hence
    C = -D
    Or A B = - B A
    This shows that the vector product is not commutative. It is anti commutative.
  2. A A = 0, i. e. the vector product of a vector by itself is zero. This is because, in this case, θ = 0, and hence sin θ = 0.
    Therefore A A = AA sin θ = 0
    Hence, the condition for two vectors to be parallel (θ = 0°) or anti parallel (θ = 180°) is that their vector product should be zero.
    If A B = 0, it means either (i) A is zero or, (ii) B = 0 or (iii) the angle θ between them is 0° or 180° .
  3. The distributive law holds for both scalar and vector products, i. e.
    A . (B + C) = A . B + A . C
    A (B + C) = A B + A C
  4. (λ A) B = A (λ B) = λ (A B); λ a real number.
  5. |A B|2 = |A|2 |B|2 = (A . B)2
  6. and are the three mutually perpendicular unit vectors at the origin O and along OX, OY and OZ respectively; the right handed rule gives:
    = = ,           = =
    = = ,           = = = 0




Test Your Skills Now!
Take a Quiz now
Reviewer Name