Math 241 Calc III Grillakis Week 2
Math 241 Calc III Grillakis Week 2 Math 241
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This 9 page Class Notes was uploaded by Colin Fields on Friday September 9, 2016. The Class Notes belongs to Math 241 at University of Maryland taught by Dr. Grillakis in Fall 2016. Since its upload, it has received 77 views. For similar materials see Calculus 3 in Math at University of Maryland.
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Date Created: 09/09/16
Calc III Lec Day 4 9/7/16 Ch. 11.5 Lines in Space 1. For a vector L parallel to a line l and a starting point on line l of r 0 the position r on line l is equal to a. b. Where t is the variable or parameter that relates to L 2. This can be broken down into the x,y,z components of a. b. Where a,b,c are arbitrary constants 3. Symmetric equations of line l a. Formed by removing the parameter, t, from the parametric equations of line l above b. This is done by solving each component for t c. 4. Distance between a point and a line a. Let l be a line parallel to vector L, P b1 a the point in question not on line l, and P bo any point on line l b. The distance D is equal to c. 5. A cylinder is a special case of all points a constant distance, r away from a line, l a. If l is set to be the z axis then the equation of the cylinder is i. Calc III Lec Day 5 9/9/16 Ch. 11.6 Planes in Space Planes are determined by a point and a nonzero vector that is perpendicular to the plane N = ai + bj + ck is the vector normal to the plane Pois a fixed point contained on the plane P is any point contained on the plane 1. Equations of a plane a. For a point P on the plane i. b. Expansion of this equation gives i. c. By setting all of the constant terms to sum to d, the equation becomes i. d. Both b and c are equations for the plane 2. Distance between a point P and 1he plane a. 3. Intersection of two planes a. Two planes have normal vectors n and nogiven w1th x,y,z components b. The intersection of two planes will be perpendicular to both normal vectors i. From this you can cross the normal vectors to get a vector v that is perpendicular to both normal vectors c. d. You then solve for a point on the plane by setting one of the variables in each plane’s equation equal to the same value i. This gives resulting equations of two variables (only y and z if you set x=1 for example) ii. You then isolate one of the variables and solve the systems of equations iii. This gives you point r ohich is a point on the line where the planes intersect e. You now have the vector v and initial point r whioh can be plugged into the standard equations of a line i. ii. Or the symmetric equations iii. 4. Axial Intercept Planes a. Making an equation for a plane with known points where it intersects the x,y, and z axes (points A,B,C) b. Consider A,B,C as falling on the x,y,z axes, respectively c. Make two vectors, one from A to B and the other from A to C d. Cross these vectors i. This gives you a vector normal to the plane formed by A,B,C e. Plug this back into the equation for a plane from a normal vector dotted to a point on the plane (1.b above) i. f. And plug in the coordinates form one of the points, A,B, or C, and you have an equation of the plane g. If you distribute everything this equation can be generalized to i.
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