Multivariable Calculus MATH 2110
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This 22 page Class Notes was uploaded by Mary Veum on Thursday September 17, 2015. The Class Notes belongs to MATH 2110 at University of Connecticut taught by Marius Ionescu in Fall. Since its upload, it has received 10 views. For similar materials see /class/205823/math-2110-university-of-connecticut in Mathematics (M) at University of Connecticut.
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Date Created: 09/17/15
The Dot Product Lecture 2 September 2 2009 The DOE Product The Dot Product 0 HO 1102 03gt and b ltb17 bz7 b3gt then thC dot product of a and b is the number 0 b 01b1 OQbQ C3b3 o It is also called the scalar product or inner product The DOE Product o lt21gtlt713gt o lt3721gtlt011gt The DOE Product Properties of the Dot Product 0 If C17 b andc are vectorsin V3 and 015 ascalar then 0 aaa2 9 abcabac 0 cabcabaCb 6 000 The DOE Product The angle between two vectors 9 If 0 is the angle between the vectors a and b then ab laHblcose o If 0 is the angle between the nonzero vectors a and b then 0050 7 g Mb The DOE Product Orthogonal vectors 7 o aandbareorthogonalifandonlyifab0 The DOE Product Direction Angles o The direction angles of a nonzero vector u are the angles a 6 and 39y that a makes with the positive Xayi and Ziaxes o The cosines of these direction angles are called the direction cosines of the the vector u a a 00804 4 cos 2 COS39y 3 M M M The DOE Product a Scalar projection of b onto a a compab W a Vector projection of b onto a a b projab W0 The DOE Product Work done by a constant force 0 The work done by a constant force F is F D7 where D is the displacement vector 0 Example A constant force F 72i 7 3 2k moves an object along a straight line from the point 17 07 0 to 737 27 3 Find the work done The DOE Product Lines and Planes in R3 Lecture 17 September 10 2009 lines and Planes in The vector equation of a line 0 A line L is determined when we know a point POM yo7 20 on L and the direction of L a Let V be a vector parallel to L and let r0 the position vector of P0 9 The vector equation of L is rrofv where f is the parameter lines and Planes in 0 Find the vector equation for the line trough the point Pil22 and parallel to the vector i 7 2 2k 0 Find the vector equation for the line trough the point PU 712 and parallel to the vector lt27 07 73 lines and Planes in The parametric equations of a line Ifr ltX y7zgt v ltayb7 cgt and re ltXo yo7zogt then ltX7 y7zgt ltgt fa yo 173720 fCgt o The parametric equations X qjcn y yobf z zocf lines and Planes in 1113 0 Find the parametric equations for the line trough the point P7 l 7 27 2 and parallel to the vector i 7 2 2k 0 Find the parametric equations for the line trough the point PU 712 and parallel to the vector lt27 07 73 lines and Planes in The line segment between two points 0 The line segment from r0 and r1 is given by the vector equation rf1ifr0fr1 091 lines and Planes in 1113 0 Find vector and parametric equations for the line trough 47 10 that is parallel to the line with parametric equations X 1 21 y 2 3fZ 171 0 Find the point of intersection of this new line with each of the coordinate planes lines and Planes in o A plane is determined by a point POM yo7 20 in the plane and a vector n that is orthogonal to the plane a The vector n is called a normal vector 0 If r0 is t he position vector of P0 and r then the vector equation of the plane nrnr0 lines and Planes in The Scalar equation of a plane a The scalar equation of the plane through POM yo 20 with normal vector r7 lt0 b Cgt is 0X7mbyi Y0CZZO 0 9 Example Find an equation of the plane through the point 2 1 73 with normal vector n 311 lines and Planes in 1113 The linear equation of a plane 9 By collecting the terms axbycz 10 9 Example Find an equation of a plane through the point 727 71 2 which is parallel to the plane 73X 2y z 7 lines and Planes in 1113 The angle between two planes a Two planes are parallel if their normal vectors are parallel o If two planes are not parallel then they intersect in a straight line and the angle between them is the acute angle between their normal vectors lines and Planes in 0 Find the angle between the planes X 7 y Z 1 and 2X y 7 3 1 9 Find a equation for the line of intersection L of these two planes lines and Planes in More Examples 0 Find a equation of a plane containing the three points 7220 7137 1 and 737 732 0 Find an equation of a plane through the point 07 4 l which is orthogonal to the line X l f y 2 7 31 Z 5 21 in which the coef cient of X is 5 0 Find an equation of a plane containing the line r lt727 72 l 1 lt7407 l which is parallel to the plane 72X2yz 5 lines and Planes in 1113
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