In vector product of vectors, the vector components are combined to give a vector. The vector product of two vectors is a vector which is perpendicular to both the given vectors. The vector product is mostly used in Physics. The vector product is also known as “cross product”.
The mathematical definition of vector product of two vectors a and b is denoted by axb and is defined as follows.
axb = |a| |b| Sin θ, where θ is the angle between a and b.
Properties of vector product:
1) axb is a vector.
2) From the definition of vector product, we have Sin θ = |axb|/|a| |b|
3) a and b are parallel if axb =0.
4) axb = -bxa
5) ixi=jxj=kxk=0, where i,j and k are mutually orthogonal unit vectors along x,y and z-axes.
6) Cross product is distributive over addition.
i.e ax(b+c) = axb+bxc
7) For any vector m, (ma)xb = m(axb)
8) If a = a1i + a2j + a3k and b = b1i + b2j + b3k then axb can be expressed as a determinant as follows.
axb = 
(This is because ixj = k, j.k= i and k.i=j)
Example:
If a = 3i + 2j + 5k and b = 2i – 6j + 4k then find axb
Solution:
By property number 8,
axb = 
= i ( 8-4) – j (12-10) + k (-18 -4) = 4i =2j -22k
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