Vector (or Cross) Product of Two Vectors

Vectors can be multiplied in two ways, a scalar product where the result is a scalar and cross or vector product where is the result is a vector. In this article, we will look at the cross or vector product of two vectors.

Explanation

We have already studied the three-dimensional right-handed rectangular coordinate system. As shown in the figure below, when the positive x-axis is rotated counter-clockwise into the positive y-axis, then a right-handed standard screw moves in the direction of the positive z-axis.

As can be seen above, in a three-dimensional right-handed rectangular coordinate system, the thumb of the right-hand points in the direction of the positive z-axis when the fingers are curled from the positive x-axis towards the positive y-axis.

Definition     

The cross or vector product of two non-zero vectors a  and b , is

a  x b  = |a | |b sinθn^

Where θ is the angle between a  and b , 0 ≤ θ ≤ π. Also, n^ is a unit vector perpendicular to both a  and b  such that a b , and n^ form a right-handed system as shown below.

As can be seen above, when the system is rotated from a  to b , it moves in the direction of n^. Also, if either a  = 0 or b  = 0, then θ is not defined and we can say,

a  x b  = 0 

Important Observations

  • a  x b  is a vector.
  • If a  and b  are two non-zero vectors, then a  x b  = 0, if and only if a  and b  are parallel (or collinear) to each other, i.e.

a  x b  = 0  a   b 

Hence, a  x a  = 0 and a  x (−a)→ = 0. This is because in the first case θ = 0. Also, in the second case θ = π, giving the value of sinθ = 0.

  • If θ = π2, then a  x b  = |a | |b |
  • Considering observations 2 and 3 above, for mutually perpendicular vectors i j , and k , we have

  • The angle between the two vectors a  and b  is,

sinθ = |a ×b ||a ||b |

  • A cross or vector product is not commutative. We know this because a  x b  = b x a . Now, we know that,

a  x b  = |a | |b sinθn^.

Where a b , and n^ form a right-handed system. Or, θ is traversed from a  to b . On the other hand,

b  x a  = |b | |a sinθn1^.

Where b a , and n1^ form a right-handed system. Or, θ is traversed from b  to a . So, if a  and b  lie on a plane of paper, then n^ and n1^ are both perpendicular to the plane of the paper. However, n^ is directed above the paper and n1^ is directed below it. Or, n^ = – n1^. Hence,

a  x b  = |a | |b sinθn^ = – |a ||b |sinθ n1^

= – b  x a 

  • From the observations 4 and 6 above, we have

j x i = – k
k x j^ = – i
i x k = – j^

  • If a  and b  represent the two sides of a triangle, then its area is |a  x b |. To understand this, look at the figure given below.

 

By the definition of the area of a triangle, we have area of ΔABC =  (AB).(CD). We know that, AB = |b | and CD = |a |sinθ. Therefore,

  • If a  and b  represent the two adjacent sides of a parallelogram, then its area is |a  x b |. To understand this, look at the figure given below.

By the definition of the area of a parallelogram, we have area of parallelogram ABCD = (AB).(DE). We know that, AB = |b | and DE = |a |sinθ. Therefore,

Area of parallelogram ABCD = |a ||b |sinθ = |a  x b |

Property: Distributivity of a cross or vector product over addition

If a b , and c  are any three vectors and λ is a scalar, then

  • a  x (b  + c ) = a  x b  + a  x c 
  • λ(a  x b ) = (λa ) x b  = a  x (λb )

Question 1: Find the area of the parallelogram whose adjacent sides are determined by the following vectors,

  • a  = i^ – j^ + 3k  and
  • b  = 2i^ – 7j^ + k .

Answer : We know that if a  and b  represent the two adjacent sides of a parallelogram, then its area is |a  x b |. Also,

Substituting the values of a1,a2,a3,b1,b2,and b3, we get

Solving the determinant, we get

  • a  x b  = {[(-1) x 1)] – [(-7) x 3]} – {[1 x 1)] – [2 x 3]} + {[1 x (-7))] – [2 x (-1)]}
    = 20
    i^ + 5j^ – 5k^.

Also, the magnitude of a  x b  is,

  • |a  x b | = [20+5+(−5)]450 25×9×2 = 152.

Therefore, the area of the parallelogram is 152.

Question : Explain the characteristics of vector product?

Answer: The characteristics of vector product are as follows:

  • Vector product two vectors always happen to be a vector.
  • Vector product of two vectors happens to be noncommutative.
  • Vector product is in accordance with the distributive law of multiplication.
  • If a • b = 0 and a ≠ o, b ≠ o, then the two vectors shall be parallel to each other.