Vector Multiplication

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Vector Multiplication: -  Two types of multiplication exist. Vector and scalar products.

The scalar product: -  This is also known as the dot product or the inner product of two vectors-

In the figure the constant force f; which acts through O. if F moved along OA, then we

Force and Displacement

Can calculate the work done by the force. Remember work done is the component of force done in the moving direction. F_h = |F| cosθ

So work done is equal to 

                                                F.a = |F|.|a|.cosθ

The scalar product of two vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is written as 

                                       

                                           a.b = a₁b₁ + a₂b₂ + a₃b_{3      (it is called a dot b)}

Geometrical form                 a.b = |a||b| cosθ

Points regarding scalar products: -

i.            The scalar product of two vectors gives a number. (a scalar)

ii.          The scalar product is a product of two vectors.

iii.       Use of dot is essential to indicate that the calculation is a scalar product.

Perpendicular vectors: -
If vectors a and b are perpendicular to each other than θ =  90

                                           a.b = |a|.|b| Cos 90   ---> a.b = 0 (we know Cos 90 = 0)

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** If a.b = 0; that does not imply that a will always perpendicular to be b. it could be a or b = 0.

Suppose a.b = b.c à a.(b – c) = 0 that does not mean that     

                       b = c. that implies that a will be perpendicular to (b – c) or a = 0.

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     For unit vectors      i.j = j.k = k.i = 0

Parallel vectors: - If vectors a and b are parallel to each other than θ = 0

                                    a.b = |a|.|b| cos 0  ---> a.b = |a|.|b|  (We know Cos 0 = 1)

Properties of scalar product: -

i.                    Commutative law:       a.b   =   b.a

ii.                 Associative law Scalar product of three vectors is not possible.

iii.               Distributive law

a.(λ.b) = (λ.a).b = λ.(a.b)           à a scalar multiplication

                  a.(b + c) = a.b + a.c             à vector addition

The vector (or cross) product: - The main practical use of the vector product is to calculate the momentum of a force in three dimensions. It is of very limited use in two dimensions.

                For Vector Cross Product Click Next

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