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Vector Addition
                                    
                                          −
                                             +
                                      =
                         +
                             −
              Given  a =  3i 2j 7k and b 2i 4j 6k
                  
                 +
                ab    =  adding the components of the three
                        areas seperately
                      = (3 + 2)i + (2 – 4)j + (–7 + 6)k
                  
                 +
                ab = 5i – 2j – k
              Vector Subtraction
              Vector subtraction can be described as the addition of a vector with the negative of another vector. For example:
              a = (3, 2)     b = (2, 1)
              a – b =     a + (–b)
              –b =     –(2, 1) = –2, –1
              a – b =     (3, 2) + (–2, –1)
              = (3 – 2) , (2 – 1) = (1, 1) = Unit Vector


              Vector Multiplication
              Before understanding vector multiplication, we need to understand that there are 2 different kinds of physical
              quantities:

                 • Scalar: Scalar quantities are those physical quantities which contains only magnitude but no direction.
                 • Vector: These quantities have both magnitude as well as direction.
              Vector multiplication is of two types: Dot product and cross product.


              Vector Dot Product
              Denoted by a,b, consider:

              a = (1, 2, 3)     b = (–4, 5, –1)
              Multiply the respective areas
              a.b = 1(–4) + 2(5) + 3(–1)

              = –4 + 10 – 3 = 3 (Dot Product is a Scalar Value.)

              Vector Cross Product
              Denoted by a x b, consider:

                    a = (3, –3, 1)     b = (–1, –4, 2)                              = (–3 × 2 –(–4 × 1) i     –(3 × 2 –
                          i   j  k    Representing        −31      3  1    3   −3      (–1 × 1)j + (3 × –4) – (–3 × –1)k
                    a b=  3 − 3 1     the three axes in   i  −42  j −  −12  + k  −1  −4  = (–6 + 4)i – (6 + 1) j + (–12 –3)k
                     ×
                         −  1 −  42   determinant form.                             = –2i – 7j – 15k
              Cross Product is a vector value.



                          Brainy Fact


                 The goal of most ML projects is to create a model to perform certain functions, such as classifying text, predicting
                 house prices, or identifying emotions. In the deep learning model, this is achieved through a neural network,
                 where the layers of the neural network use matrix and vector multiplication to adjust parameters.






                    170     Touchpad Artificial Intelligence (Ver. 2.0)-XI
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