In scalar product of vectors, the vector components are combined to give a scalar. The scalar product of two vectors is the product of the component of one vector (in the direction of the other vector) and the second vector. The scalar product is mostly used in Physics. The scalar product is also known as “dot product”.
The mathematical definition of scalar product of two vectors a and b is denoted by a.b and is defined as follows.
a.b = |a| |b| Cos θ, where θ is the angle between a and b.
Properties of scalar product:
1) a.b is a scalar.
2) From the definition of scalar product, we have Cos θ = a.b/|a| |b|
3) a and b are orthogonal if a.b =0.
4) a.b = b.a
5) i.i=j.j=k.k=1, where i,j and k are mutually orthogonal unit vectors along x,y and z-axes.
6) Dot product is distributive over addition.
i.e a.(b+c) = a.b+b.c
7) For any scalar m, (ma).b = m(a.b)
8) If a = a1i + a2j + a3k and b = b1i + b2j + b3k then a.b = a1b1 + a2b2 + a3b3.
(This is because i.j = j.k=k.i=0 because they are mutually orthogonal)
Example:
If a = 3i + 2j + 5k and b = 2i – 6j + 4k then find a.b
Solution:
By property number 8,
a.b = 3*2 + 2*-6 + 5*4 = 6 -12 + 20 = 14.
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