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lecture 1

# lecture 1 - Introduction Vectors and Integrals Vectors...

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Unformatted text preview: Introduction: Vectors and Integrals Vectors Vectors are characterized by two parameters: • length (magnitude) • direction These vectors are the same a r a r a r Sum of the vectors: b a + r r a r b r a b + r r a r b r a r b r = Vectors Sum of the vectors: for a larger number of vectors the procedure is straightforward a r b r a b c + + r r r a r b r c r c r 2 a a a + = r r r a r a r Vector (where is the positive number) has the same direction as , but its length is times larger c a r c a r a r 2 a r c Vector (where is the negative number) has the direction opposite to , and times larger length c a r c a r c a r 2 a- r usually have unit magnitude Vectors The vectors can be also characterized by a set of numbers (components), i.e. This means the following : if we introduce some basic vectors, for example x and y in the plane, then we can write 1 2 ( , ,...) a a a = r 1 2 a a x a y = + r r r y r x r a r 2 a y r 1 a x r Then the sum of the vectors is the sum of their components: 1 1 2 2 ( , ) a b a b a b + = + + r r 1 2 ( , ) a a a = r 1 2 ( , ) b b b = r 1 2 ( , ) a a a = r 1 2 ( , ) ca ca ca = r , x y r r Vectors: Scalar and Vector Product a r b r a b r r is the scalar (not vector) cos( ) ab ϕ = ϕ a b c = r r r is the VECTOR, the magnitude of which is Vector is orthogonal to the plane formed by and sin( ) ab ϕ c r a r b r a r b r ϕ c r If the vectors are orthogonal then the scalar product is 0 a r b r a b = r r Scalar Product Vector Product If the vectors have the same direction then vector product is 0 a b = r r a r b r from the definition of the scalar product Vectors: Scalar Product a r b r a b r r is the scalar (not vector) cos( ) ab ϕ = ϕ If the vectors are orthogonal then the scalar product is 0 a r b r a b = r r Scalar Product y r x r a r 2 a y r 1 a x r It is straightforward to relate the scalar product of two vectors to their components in orthogonal basis If the basis vectors are orthogonal and have unit magnitude (length) then we can take the scalar product of vector and basis vectors : 1 2 a a x a y = + r r r , x y r r 1 2 a a x a y = + r r r , x y r r 1 2 1 cos( ) a a x a x x a y x a ϕ = = + = r r r r r r ϕ =0 (orthogonal)...
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lecture 1 - Introduction Vectors and Integrals Vectors...

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