The vector product of two vectors a and b is given by a vector whose magnitude is given by |a||b|sinθ (where0∘≤θ≤180∘), which stands for the angle between the two vectors. Note that the direction of the resultant vector is denoted by a unit vector ^n whose direction is perpendicular to both the vectors a and b in a way that a, b and ^n are oriented in right-handed system.
Right-handed orientation happens when vector a is twisted in the direction of vector b, then the direction of the unit vector ^n goes in the direction in which a right-handed screw would spin if moved in a similar manner. Also, these given vectors a and b cannot be called null vectors or non-parallel in nature. The right-hand thumb rule gives a clear picture of the direction of the resultant vector.
Therefore, we can conclude that, a×b = |a||b|sinθ ^n, where a×b stands for the cross product of two vectors. In any given situation, if the vectors are null or both the vectors are parallel to each other, then the cross product cannot be defined.
In this case, we can conclude that a×b = 0
Let’s assume that a and b are the adjacent sides of the parallelogram OACB and the angle between the vectors a and b is θ. Then, we can say that the area of the parallelogram is denoted by |a×b| = |a||b|sinθ.
Properties of Cross Product of Two Vectors
i) The vector product never has a Commutative Property. It is denoted by,
a×b = – (b×a)
ii) The property given below is true in the case of vector multiplication:
(ka)×b= k(a×b) =a×(kb)
iii) If the vectors mentioned are collinear then
(Since the angle between both the vectors would be 0, then sin 0 = 0)
iv) As per the above property
We can conclude that the vector multiplication of a vector with itself would be
a×a= |a||a|sin0 ^n = 0
Also, when it comes to unit vector notation
As per the above discussion
This example can be explained better with the help of the following diagram. When moving in clockwise direction and considering the cross product of any two pair of the unit vectors, we can derive the third one and we get the negativeresultant in anticlockwise direction.
v) a × b in terms of unit vectors can be represented as
Then →a×→b =(a1^i+a2^j+a3^k)(b1^i+b2^j+b3^k)
When expanded, we would get
|a||b|sinθ ^n = (a2b3–a3b2)^i+(a3b1–a1b3)^j+(a1b2–a2b1)^k
vi) Distributive Law: a×(b+c) = a×b+a×c