Chapter 3 Vectors I Definition II Arithmetic operations involving vectors A Addition and subtraction Graphical method Analytical method Vector components B Multiplication Review of angle reference system 90 0 1 90 90 2 180 2 1 180 3 180 3 270 0 Origin of angle reference system 4 270 4 360 270 Angle origin 4 300 60 I Definition Vector quantity quantity with a magnitude and a direction It can be represented by a vector Examples displacement velocity acceleration Same displacement Displacement does not describe the object s path Scalar quantity quantity with magnitude no direction Examples temperature pressure II Arithmetic operations involving vectors Vector addition s a b Geometrical method a b s a b Rules a b b a commutative law a b c a b c 3 1 associative law 3 2 Vector subtraction d a b a b 3 3 Vector component projection of the vector on an axis a x a cos 3 4 a y a sin a a x2 a 2y Scalar components of a Vector magnitude 3 5 tan ay ax Vector direction Unit vector Vector with magnitude 1 No dimensions no units i j k unit vectors in positive direction of x y z axes a a x i a y j 3 6 Vector component Vector addition Analytical method adding vectors by components r a b a x bx i a y by j 3 7 Vectors Physics The relationships among vectors do not depend on the location of the origin of the coordinate system or on the orientation of the axes The laws of physics are independent of the choice of coordinate system a a x2 a 2y a 2x a 2y 3 8 Multiplying vectors Vector by a scalar f s a Vector by a vector Scalar product scalar quantity dot product a b ab cos a x bx a y by a z bz 3 9 Rule a b b a a b ab cos 1 0 a b 0 cos 0 90 3 10 i i j j k k 1 1 cos 0 1 i j j i i k k i j k k j 1 1 cos 90 0 a b cos a b Angle between two vectors Multiplying vectors Vector by a vector Vector product vector cross product a b c a y bz by a z i bz a x a z bx j a x by bx a y k c ab sin Magnitude a b 0 sin 0 0 a b ab sin 1 90 Vector product Direction right hand rule Rule b a a b 3 12 c perpendicular to plane containing a b 1 Place a and b tail to tail without altering their orientations 2 c will be along a line perpendicular to the plane that contains a and b where they meet 3 Sweep a into b through the smallest angle between them Right handed coordinate system z k i j y x Left handed coordinate system z k j y i x i i j j k k 1 1 sin 0 0 i i j j k k 0 i j j i k j k k j i k i i k j the result is a vector in the positive direction of the y axis with a P1 If B is added to C 3i 4j magnitude equal to that of C What is the magnitude of B Method 2 Method 1 Isosceles triangle B C B 3i 4 j D D j C D 32 4 2 5 B 3i 4 j 5 j B 3i j B 9 1 3 2 C D tan 3 4 36 9 B 2 sin B 2 D sin 3 2 D 2 2 B P2 A fire ant goes through three displacements along level ground d1 for 0 4m SW d2 0 5m E d3 0 6m at 60 North of East Let the positive x direction be East and the positive y direction be North a What are the x and y components of d1 d2 and d3 b What are the x and the y components the magnitude and the direction of the ant s net displacement c If the ant is to return directly to the starting point how far and in what direction should it move a d1x 0 4 cos 45 0 28m N D 45 d4 d3 d1 E d1 y 0 4 sin 45 0 28m d 2 x 0 5m d2 y 0 d 3 x 0 6 cos 60 0 30m d2 d 3 y 0 6 sin 60 0 52m b d 4 d1 d 2 0 28i 0 28 j 0 5i 0 22i 0 28 j m D d 4 d 3 0 22i 0 28 j 0 3i 0 52 j 0 52i 0 24 j m D 0 52 2 0 24 2 0 57m 0 24 24 8 North of East 0 52 tan 1 c Return vector negative of net displacement D 0 57m directed 25 South of West P2 d1 4i 5 j 6k d 2 i 2 j 3k d 4i 3 j 2k 3 a r d1 d 2 d 3 b Angle between r and z c Component of d1 along d 2 d Component of d1 perpendicular to d 2 and in plane of d1 d 2 a r d1 d 2 d 3 4i 5 j 6k i 2 j 3k 4i 3 j 2k 9i 6 j 7k 7 b r k r 1 cos 7 cos 1 123 12 88 d1perp r 9 2 6 2 7 2 12 88m d1 d 2 c d1 d 2 4 10 18 12 d1d 2 cos cos d1d 2 d d 12 d1 d1 cos d1 1 2 3 2m d1d 2 3 74 d1 d1 d2 d 2 12 2 2 32 3 74m d d1 d12 d12perp d1 perp 8 77 2 3 2 2 8 16m d1 4 2 52 6 2 8 77 m P3 If d1 3i 2 j 4k d 2 5i 2 j k Tip d1 d 2 d1 4d 2 Think before calculate d1 d 2 a contained in d1 d 2 plane d1 4d 2 4 d1 d 2 4b perpendicular to d1 d 2 plane a perpendicular to b cos 90 0 4a b 0 P4 Vectors A and B lie in an xy plane A has a magnitude 8 00 and angle 130 B has components Bx 7 72 By 9 20 What are the angles between the negative direction of the y axis and a the direction of A b the direction of AxB c the direction of Ax B 3k y A a Angle between y and A 90 50 140 130 x B b Angle y A B C angle j k because C perpendicular plane A B xy 90 c Direction A B 3k D E B 3k 7 72i 9 2 j 3k i D A E 5 14 j 6 13 k 0 18 39i 15 42 j 94 61k 7 72 9 …
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