vector revision

# Subtraction is a b k i j a a b b ab ab c b a bc c

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Subtraction is a + ( b ) k i j a a-b b a+b a+b c b a b+c c

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Properties of addition/subtraction 1.9 The following results follow immediately from the above definition of vector addition (incl. subtraction). 1. a + b = b + a (it commutes) 2. ( a + b ) + c = a + ( b + c ) = a + b + c (it associates) 3. a + 0 = 0 + a = a where the zero vector is 0 = [0 , 0 , 0]. 4. a + (- a ) = 0
Multiplication of a vector by a scalar 1.10 NOT the scalar product! Just as for matrices, multiplication of a vector a by a scalar c is defined as multiplication of each component by c , so that c a = [ ca 1 , ca 2 , ca 3 ] . It follows that: | c a | = radicalbig ( ca 1 ) 2 + ( ca 2 ) 2 + ( ca 3 ) 2 = | c || a | . The direction of the vector will reverse if c is negative, but otherwise is unaffected. A vector where the sign is uncertain is called a director .

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Example 1.11 The electrostatic force on charged particle Q due to another charged particle q 1 is F = K Qq 1 r 2 ˆ r where constant K = 1 4 πǫ r ǫ 0 where r is the vector from q to Q . Question: Write down an expression for the force on Q at R due to N charges q i at r i , i = 1 , . . . , N . Answer: The vector from q i to Q is R r i . The unit vector in that direction is ( R r i ) / | R r i | , so F ( r ) = N summationdisplay i =1 K Qq i | R r i | 3 ( R r i ) q i r i R Q Notice that we are thinking algebraically about vectors — not fussing about their components. Not a coordinate system in sight.
Scalar product or dot product 1.12 The scalar product of two vectors results in a scalar quantity: a · b = a 1 b 1 + a 2 b 2 + a 3 b 3 . Note that a · a = a 2 1 + a 2 2 + a 2 3 = | a | 2 = a 2 . These properties of the sprod follow immediately: – a · b = b · a (it commutes) – a · ( b + c ) = a · b + a · c (it distributes w.r.t vector addition) ( λ a ) · b = λ ( a · b ) = a · ( λ b ) (scalar multiple of a scalar product of two vectors)

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Geometrical interpretation of scalar product 1.13 Consider the square magnitude of the vector ( a b ). | a b | 2 = ( a b ) · ( a b ) = a · a + b · b 2( a · b ) = a 2 + b 2 2( a · b ) The cosinerule says length AB 2 is | a b | 2 = a 2 + b 2 2 ab cos θ Hence a · b = ab cos θ, independent of the coord system. Conversely cos θ = a · b /ab θ b B a-b a A O
Projection of one vector onto the other 1.14 θ b direction of a Projection of b onto a b cos θ is the component of b in the direction of a . a cos θ is the component of a in the direction of b . Projection is v. useful when the second vector is a unit vector. a · ˆ ı is the size of the component of a in the direction of ˆ ı . To get the vector component of b in the dirn of a ( b · ˆ a ) ˆ a = 1 a 2 ( b · a ) a . So ( a · ˆ ı ı is the vector component of a in the direction of ˆ ı .

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Orthonormal vectors and coordinates 1.15 In the particular case a · b = 0, the angle between the two vectors is a right angle. The vectors are said to be orthogonal — neither has a component in the direction of the other.
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