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CE Statics Lecture 6

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Contents Dot Product

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DOT PRODUCT Dot product is the method of multiplying two vectors and is used to solve three-dimensional problems If A and B are two vectors then, A . B = AB cos () where 0 180 A B

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**Laws of Operation (1) Commutative Law A . B = B . A**

(2) Multiplication by a scalar a ( A . B ) = ( aA ) . B = A . ( aB ) = ( A . B ) a (3) Distribution Law A . ( B + D ) = ( A . B ) + ( A . D )

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**Cartesian Vector Formulation**

i . i = 1 j . j = 1 k . k = 1 i . j = 0 i . k = 0 k . j = 0 i . i = (1) (1) cos 0 = 1 i . k = (1) (1) cos 90 = 0 If A and B are Cartesian vectors, then A . B = (Ax i + Ay j + Az k) . (Bx i + By j + Bz k = Ax Bx (i.i) + Ax By (i.j) + Ax Bz (i.k) Ay Bx (j.i) + Ay By (j.j) + Ay Bz (j.k) Az Bx (k.i) + Az By (k.j) + Az Bz (k.k) Doing Dot Product, we have A . B = Ax Bx + Ay By + Az Bz [scalar ( so there is no i, j, k)]

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Applications (1) to find the angle formed between two vectors or two intersecting lines. If A and B are vectors, then between their tails will be: = cos-1 ( A . B ) / (AB) 180 If A . B = = cos-1 (0) = 90 (A is perpendicular to B)

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**to find the rectangular components of a vector. AII = A cos () **

AII = A cos = A . u AII = A cos () u = (A . u) u AI = A – AII (since A = AII + AI) To find the magnitude of AI If cos () = A . U /A = cos-1 ( A . u ) / A (0 180) AI = A sin () or AI = A2 – (AII)2 (Pythagorean) a a AII = A cos () u AI A u

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Examples Examples 2.16 – 2.17 Problem 2.113 Problem 2.129

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**Assignment No. 1 (Chapter 2)**

1, 9, 12, 19, 23, 28, 29, 33, 36, 40, 47, 54, 58, 60, 65, 69, 73, 76, 78, 80

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