Product of a vector
We know that vectors can be added and substracted, Likewise vectors can be multiplied to get another result. When we were learning addition we have seen that two can vectors can be added only when they are of same type ,but in case of product it is not necessary for vectors to be same. We can multiply any physical quantity .There are two method of multiplying vectors.
1) Scalar Product (Dot Product) : The scalar product or dot product of two non zero vector is defined as product of magnitude of two vector and a cosine of angle θ between them.
If P and Q are the two vectors then
P . Q = PQcosθ
dot sign is used between the vectors, hence it is called as dot product.
Conclusion
=P (Q cosθ)
=P( Component of Q in direction of P)
Similarly,
=Q (P cosθ)
Different cases
In the scalar product
i) If θ = 0
î . î = ĵ . ĵ = k̂ . k̂ = 1
ii) If θ = 90
î . ĵ = ĵ . k̂ = k̂. î = 0
iii) If θ = 180
Proof for scalar multiplication of two vector
Now,
= (Px î + Py ĵ + Pz k̂) . (Qx î + Qy ĵ + Qz k̂)
= (Px î ) . ((Qx î + Qy ĵ + Qz k̂) +
(Py ĵ). (Qx î + Qy ĵ + Qz k̂)+
(Pz k̂).(Qx î + Qy ĵ + Qz k̂)
= Px Qx (î .î ) + Px Qy (î .ĵ) + Px Qz (î .k̂) +
Py Qx (ĵ .î ) + Py Qy (ĵ .ĵ) + Py Qz (ĵ .k̂) +
Pz Qx (k̂ .î ) + Pz Qy (k̂ .ĵ) + Pz Qz (k̂ .k̂)
we know ,
î . î = ĵ . ĵ = k̂ . k̂ = 1
î . ĵ = ĵ . k̂ = k̂. î = 0
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