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vector revision

To get the vector component of b in the dirn of a b a

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Unformatted text preview: To get the vector component of b in the dirn of a ( b a ) a = 1 a 2 ( b a ) a . So ( a ) is the vector component of a in the direction of . 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. In 3D, an orthogonal coordinate system is characterised by = = k k = 1 and = k = k = 0 A scalar product is an inner product 1.16 We have been writing vectors as row vectors a = [ a 1 , a 2 , a 3 ] Its convenient: it takes less space than writing column vectors In matrix algebra, vectors are column vectors . So, M a = v means M 11 M 12 M 13 M 21 M 22 M 23 M 31 M 32 M 33 a 1 a 2 a 3 = v 1 v 2 v 3 and row vectors are written as a ( a transpose). Most times can be relaxed, but need to fuss to point out that the scalar product is also the inner product used in linear algebra. The inner product is defined as a b a b = [ a 1 , a 2 , a 3 ] b 1 b 2 b 3 = a 1 b 1 + a 2 b 2 + a 3 b 3 = a b Scalar Product Example 1 1.17 Question A force F is applied to an object as it moves by a small amount r . What work is done on the object by the force? Answer The work done is equal to the component of force in the direction of the displacement multiplied by the displacement itself. This is just a scalar product: W = F r . Later we will see how to integrate such elements over particular paths as line integrals . Scalar Product Example 2 1.18 Question A cube has four diagonals, connecting opposite vertices. What is the angle between an adjacent pair? Answer k i j [-1,1,1] [1,1,1] The directions of the diagonals are [ 1 , 1 , 1]. The ones shown in the figure are [1 , 1 , 1] and [ 1 , 1 , 1]. The angle is thus = cos 1 [1 , 1 , 1] [ 1 , 1 , 1] 1 2 + 1 2 + 1 2 1 2 + 1 2 + 1 2 = cos 1 (1 / 3) Scalar Product Example 3 1.19 Question: Pinball with velocity s bounces (elastically) from a baffle whose endpoints are p and q . What is the velocity vector after the bounce? Answer ^ ^ p v u q s Refer to coord frame with principal directions along and perpendicular to the baffle: u = [ u x , u y ] = q p | q p | v = u = [ u y , u x ] Before impact: velocity is s before = ( s . u ) u + ( s . v ) v After impact: component of velocity in dirn of baffle u is same . component normal to the baffle along v is reversed s after = ( s . u ) u ( s . v ) v Scalar Product Example 3 1.20 Worth reflecting on this example ......
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To get the vector component of b in the dirn of a b a a = 1...

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