1. Possible products of two vectors - vector and scalar, definition and analytical formula. (4p.)

Question 1

The scalar product

Answer 1

Let’s assume that:

𝐴 = 𝑎 , 𝑎 ! , 𝑎 !

𝐵 = 𝑏 ! ! , 𝑏 ! , 𝑏 !

The scalar product (a.k.a. dot product) of these two vectors can be defined algebraically:

𝐴 ∙ 𝐵 = 𝑎 ! 𝑏 ! + 𝑎 ! 𝑏 ! + 𝑎 ! 𝑏!

and geometrically:

𝐴 ∙ 𝐵 = 𝐴 𝐵 cos 𝛼

where 𝛼 is the angle between these two vectors. As the name suggests, the result of this operation is a scalar (number without directional properties).

Question 2

The cross product

Answer 2

The cross product of these two vectors can be defined in the following way: 𝐴×𝐵 = 𝐶 The coordinates of resultant vector 𝐶 can be obtained by calculating the determinant of the following matrix:

𝚤 𝚥 𝑘

= 𝑎 𝑏 − 𝑎! ! 𝑏 𝚤 𝑎 𝑏 − 𝑎 𝑏 𝑏 ! − 𝑎 ! 𝑏 ! 𝑘 𝑎 ! 𝑎 𝑎 ! 𝑎! ! ! ! ! + ! ! ! ! 𝚥 +

𝑏 ! 𝑏 ! 𝑏 !

therefore 𝐶 = [ 𝑎 ! 𝑏 ! − 𝑎 ! 𝑏 ! , 𝑎! 𝑏 − 𝑎 ! 𝑏 ! , 𝑎 ! 𝑏 ! − 𝑎 ! 𝑏 ! ] .

Geometrically, cross product is defined as:

𝐴×𝐵 = 𝐴 𝐵 sin 𝛼 It is important to remember that cross product is anticommutative 𝐴×𝐵 = − ( 𝐵×𝐴 )