Multiplication of Vectors


1. Multiplication of vector by a scalar 

Let vector a is multiplied by a scalar m. If m is a positive quantity, only magnitude of the vector will change by a factor ‘m’ and its direction will remain same. If m is a negative quantity the direction of the vector will be reversed. 

2. Multiplication of a vector by a vector 
  
   (i) Dot product or scalar product 
   (ii) Cross product or vector product 

Dot product or scalar product 

   The dot product of two vectors a and b is defined as 
                              a-> . b->  = ab cosθ 

where a and b are the magnitudes of the respective vectors and θ is the angle between them. The final product is a scalar quantity. If two vectors are mutually perpendicular then θ = 900 and cos 90 = 0, Hence, their dot product is zero. 

   Some examples of dot product: work = F-> . s->  = Fs cosθ

                                      dot-product  

Here, 
              mathematical-dot-product

The dot product obeys commutative law 

      i.e.     a ->. b-> =b-> .a->  

Hence,   a-> . b->  =  axbx + ayby + azbz 

Illustration : 

Find the angle between the vectors A and B where 

                                mathematical-illustration                   
Solution : 

We know 

              A->.B->  = |A||B| cosθ   where |A| = √(22 +32 + 32) = √22,
 

  |B| =√(12 + 22+ 32) = √14 

Hence cosθ = (A-> .B-> )/(|A||B|)=((2i +3j +3k  )(i +2j -3k ))/(√22×√14)
                       
                          =(2+6-9)/(2√77)=(-1)/(2√77) 

                             => θ = cos-1((-1)/(2√77)) 

Cross product or vector product 

          The cross product of the two vectors a and b is defined as 

                                 a->  × b->  = c->  

     Here, |c-> |=|a->|×|b->| sinθ, where θ is the angle between the vectors. 

Vector product is defined as a vector quantity with a direction perpendicular to the plane containing vectors A and B then C = AB sin θ n, where n is a unit vector perpendicular to the plane of vector A and vector B. To specify the sense of the vector C, refer to the figure given below.

                                cross-product 

Imagine rotating a right hand screw whose axis is perpendicular to the plane formed by vectors A and B so as to turn it from vectors A to B trough the angle θ between them. Then the direction of advancement of the screw gives the direction of the vector product vectors A-> × B->

Illustration: 

Obtain a unit vector perpendicular to the two vectors A-> = 2i + 3j + 3k, B-> = i – 2j + 3k 

Solution: 

We know that A-> × B-> = AB sin θ n  

                                     n  =  (A->×B->)/ABsinθ 

We have A-> × B->  = 17i-2j-7k  

                                      A = √29 B = √14 

and θ = cos-1 8/(√14 √29) (Use concept of dot product to find θ). 

From the above values we can find n 

Solving we get, 

               n = (17i-2j-7k)/(√29 √14 sinθ)     where θ cos-1 8/(√14 √29)

Cross Product of Parallel vectors 

If two vectors are parallel or antiparallel, then θ is either 00 or 1800. Since sin 00and sin 1800 both equals zero. Hence magnitude of their cross product is zero. 

The vector product does not follow commutative law.

                                         mathematical-cross-product 

Product of unit vectors

                                              product-of-unit-vectors 


           mathematical-product-of-unit-vectors

Multiplication of vectors is a very important topic from IIT JEE, AIEEE and other engineering exams perspective. Dot product or scalar product and cross product or vector product find their use both in physics and mathematics while preparing for IIT JEE, AIEEE and other engineering entrance exams. 

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