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Math 232 Fall 2002 Write v^!^ !" v 1 (^) , v 2 (^) , v 3 #and 0^! !" 0, 0, 0#. Define v $ w !" v 1 (^) , v (^) 2 , v (^) 3 # $ " w 1 (^) , w 2 (^) , w 3 (^) #!" v 1 (^) $ w 1 (^) , v 2 (^) $ w 2 (^) , v 3 (^) $ w 3 # !! (^) and tv^!^! t " v 1 (^) , v 2 (^) , v 3 (^) #!" tv 1 (^) , tv 2 (^) , tv 3 #. The dot product is v^!^ % w!^! v w 1 1 (^) & v w 2 2 (^) & v w 3 3. The length is v^!^! v!^ % v!^. With t! 1/ v!^ , u!^! tv !has length 1. Let v^!^ and w^!^ span two sides of a triangle. Represent the third side by w^!^ ' v!^. Let the angle between v^!^ and w^!^ be given by_!. According to the Law of Cosines, it must be that w^!^ ' v!^2! w!^^2 & v!^2 ' 2 w!^ v!^ cos!. Expand and conclude that v^!^ % w!^! v!^ w!^ cos!_. The cross product is v^!^ ( w!^ !" v w 2 3 (^) ' v w 3 2 (^) , v w 3 1 (^) ' v w 1 (^) 3 , v w 1 2 (^) ' v w 2 1 #. Inspection reveals that w^!^ ( v!^! ' v!^ ( w!^ and v^!^ % ( v!^ ( w!^ )! w!^ % ( v!^ ( w!^ )! 0. The length of the cross product satisfies ) * ) * ) * ) * ) * ) * (^) ) *
(^2 2 3 3 2 2 3 1 1 3 2 1 2 2 )
1 2 3 2 1 2 3 2 1 1 2 2 3 3 2 2 2 2.
v w v w v w v w v w v w v w v v v w w w v w v w v w v w v w
In terms of the angle_!_ between v^!^ and w^!^ this is v^!^ ( w!^! v!^ w!^ sin_!. Application: Given two points P and Q let the vector p^!^ and the vector q^!^ go from the origin to P and Q respectively. Let v^!^! q!^ ' p!^. The parametric curve r t !( )^! p!^ & tv! traces a straight line, which passes through P when t! 0 and Q when t! 1. The midpoint of the straight line segment between P and Q is r! ( )^12! p!^ & 12 v^!! 12 ( p!^ & q !)^. If there is a mass mP at P and a mass mQ at Q , then the center of mass is (^) m (^) P &^1 m (^) Q ( m pP^!^ & m qQ!^ ). Application: Let P! ( p 1 (^) , p 2 (^) , p 3 )and Q! ( q 1 (^) , q 2 (^) , q 3 ). Connect P , Q so that v^!^ !" q 1 (^) ' p 1 (^) , q (^) 2 ' p 2 (^) , q (^) 3 ' p 3 #. The distance between P and Q is given by v^!^! v!^ % v!^! ( q 1 (^) ' p 1 (^) )^2 & ( q 2 (^) ' p 2 (^) )^2 & ( q (^) 3 ' p 3 )^2. All points ( , , ) x y z a distance r # 0 from P satisfy ( x ' p 1 (^) )^2 & ( y ' p 2 (^) )^2 & ( z ' p 3 )^2! r^2. This is the equation of a sphere of radius r with center at P. Application: Let!_ be the angle between v^!^ !" v 1 (^) , v 2 (^) , v 3 #and w^!^ !" w 1 (^) , w 2 (^) , w 3 #. It must be that
cos 1 (^ v^ w ) ! (^) v w ! '^ %
!!. In particular,!! " / 2 if and only if v^!^ % w!^! 0. The vectors v^!^ and w^!^ are
perpendicular if and only if v^!^ % w!^! 0. Application: Let P! ( p 1 (^) , p 2 (^) , p 3 )be some given point in a plane. Let Q! ( , , x y z )be an arbitrary point in the same plane. The vectors v^!^ !" x ' p 1 (^) , y ' p (^) 2 , z ' p 3 #are always in the plane. Let n + 0
be a fixed vector such that n!^ % v^ !! 0. Any such n^!^ is a normal vector to the plane. If n^!^ !" A B C , , #, then n!^ % v^ !! 0 is equivalent to Ax & By & Cz! Ap 1 (^) & Bp 2 (^) & Cp 3! D. Application: Let_!_ be the angle between v^!^ and w^!^. The quantity w^!^ cos_!_ is the size of the adjacent side, in
the direction of v^!^ , of a triangle with hypotenuse along w^!^. This value is u!^ % w! where u^!^ is a unit vector in the direction given by v^!^. The vector ( u!^ % w u !)!^ is the projection of w^!^ onto v^!^. Application: Let n^!^ !" A B C , , #be a normal vector to a plane Ax & By & Cz! D with p^!^ a vector pointing to a point ( p 1 (^) , p 2 (^) , p 3 )in the plane. Let u^!^ be the unit vector in the direction given by n^!^. The triangle created by p^!^ as the hypotenuse and u^!^ along one of the short sides has ( p!^ % u u! !) pointing to a point in the plane. All other points in the plane must be further away from the origin. It follows that the distance from the plane to the origin is p!^ % u != D / A^2 & B^2^ & C^2. The distance from the plane to a point ( q 1 (^) , q 2 (^) , q 3 )pointed to by q^!^ is the same formula with p^!^ ' q! replacing p^!^ so D! A p ( 1 (^) ' q 1 (^) ) & B p ( 2 (^) ' q (^) 2 ) & C p ( 3 (^) ' q 3 ). Application: Since v^!^! tw!^ if and only if v ( w! 0
, it follows that two vectors are parallel if and only if their cross product vanishes. Application: The area of the parallelogram spanned by v^!^ and w^!^ is given by v^!^ ( w!^. Application: The area of a triangle with vertices P! ( p 1 (^) , p 2 (^) , p 3 ), Q! ( q 1 (^) , q 2 (^) , q 3 ), and R! ( r 1 (^) , r 2 (^) , r 3 )is v^!^ ( w!^ / 2, where v^!^ !" q 1 (^) ' p 1 (^) , q 2 (^) ' p (^) 2 , q (^) 3 ' p 3 #and w^!^ !" r 1 (^) ' p 1 (^) , r 2 (^) ' p 2 (^) , r 3 (^) ' p 3 #represents two of its sides. Application: The parallelepiped spanned by u^!^ , v^!^ and w^!^ has volume u!^ % ( v!^ ( w !). Application: Visualize the vectors perpendicular to the line p!^ & tv !. Consider the opposite side in a right angle triangle with hypotenuse p^!^ and adjacent side tv^!^ for some t. With_!_ the angle between p^!^ and v^!^ , the length of the opposite side is p^!^ sin_!_. This value is equal to u^!^ ( p!^ where u^!^ is a unit vector in the direction given by v^!^. It follows that the distance from the line p!^ & tv! to the origin is u^!^ ( p!^. The distance from the line to some arbitrary point q^!^ is u^!^ ( ( p!^ ' q !)^. Application: Consider the two lines p!^ & tv! and q!^ & sw !. Visualize the triangle with hypotenuse p^!^ ' q!^ and adjacent side the perpendicular segment between the two lines. If the lines are parallel, then the distance between the two lines equals the distance from q^!^ to the first line. If the lines are not parallel, then the vector v ( w + 0
is perpendicular to both lines. Let_!_ be the angle between p^!^ ' q!^ and v^!^ ( w!^. The length of the opposite side of the triangle is p^!^ ' q!^ sin_!_. Let u^!^ be a unit vector in the direction given by v^!^ ( w!^. The distance between the two lines is u^!^ ( ( p!^ ' q !)^. Application: Suppose a line of direction v^!^ bounces of a surface with normal n^!^. The reflected lines direction
w^!^ is given by w! v '2(^ v^ n^ % n^ ) nn
!!!^!^!
