Vector Product
1) Vector Product (Cross Product) : The vector product or cross product of two non zero vector is defined as product of magnitude of two vector and a sine of angle θ between them.
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| Vector product |
If P and Q are the two vectors then
P x Q = PQsinθ
where 0 ≤ θ ≤ Ï€
cross sign is used between the vectors, hence it is called as cross product.
According to right hand screw rule, if the screw is rotated from P to Q then the direction of R vector is given by direction in which screw advances.
Conclusion
Different cases
In the vector product
i) If θ = 0°
Conclusion: If two non zero vectors are parallel, their cross product is zero.
ii) If θ = 180°
Conclusion: If two non zero vectors are antiparallel, their cross product is zero.
iii) If θ = 90°
Conclusion: If two non zero vectors are perpendicular, their cross product is equal to product of their magnitude.
iv) î x î = ĵ x ĵ = k̂ x k̂ = 0
v) î x ĵ = k̂ , ĵ x k̂ = î , k̂x î = ĵ
vi) ĵ x î = -k̂ , k̂ x ĵ = -î , î x k̂= -ĵ
Proof for vector multiplication of two vector
Now,
= (Px î + Py ĵ + Pz k̂) x (Qx î + Qy ĵ + Qz k̂)
= (Px î ) x ((Qx î + Qy ĵ + Qz k̂) +
(Py ĵ) x (Qx î + Qy ĵ + Qz k̂)+
(Pz k̂) x (Qx î + Qy ĵ + Qz k̂)
= Px Qx (î x î ) + Px Qy (î x ĵ) + Px Qz (î x k̂) +
Py Qx (ĵ x î ) + Py Qy (ĵ x ĵ) + Py Qz (ĵ x k̂) +
Pz Qx (k̂x î ) + Pz Qy (k̂x ĵ) + Pz Qz (k̂x k̂)
we know ,
î x î = ĵ x ĵ = k̂ x k̂ = 0
î x ĵ = k̂ , ĵ x k̂ = î , k̂x î = ĵ
ĵ x î = -k̂ , k̂ x ĵ = -î , î x k̂= -ĵ
(-î)Pz Qy+0
ĵ (Pz Qx - Px Qz ) +
k̂(Px Qy - Py Qx)
This can be written in determinant form as
Previous Topic : Scalar Product
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