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# TOP IN APPLIED MATH I MATH 311

Texas A&M
GPA 3.6

Yaroslav Vorobets

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COURSE
PROF.
Yaroslav Vorobets
TYPE
Study Guide
PAGES
24
WORDS
KARMA
50 ?

## Popular in Mathematics (M)

This 24 page Study Guide was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Study Guide belongs to MATH 311 at Texas A&M University taught by Yaroslav Vorobets in Fall. Since its upload, it has received 69 views. For similar materials see /class/226003/math-311-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15
MATH 311 504 Topics in Applied Mathematics Lecture 13 Review for Test 1 Topics for Test 1 Vectors WilliamsonTrotter 11 12 14 16 22C 0 Vector addition and scalar multiplication 0 Length of a vector angle between vectors 0 Dot product orthogonality 0 Cross product mixed triple product 0 Linear dependence Analytic geometry WilliamsonTrotter 13 15 16 0 Lines and planes parametric representation 0 Equations of a line in R2 and of a plane in R3 0 Distance from a point to a line in R2 or from a point to a plane in R3 0 Area ofa triangle and a parallelogram in R3 0 Volume of a parallelepiped in R Topics for Test 1 Systems of linear equations Wiliamson Trotter 21 22 Elimination and back substitution Elementary operations Gaussian elimination Matrix of coefficients and augmented matrix Elementary row operations Row echelon form and reduced row echelon form Free variables parametric representation of the solution set Homogeneous systems checking for linear independence of vectors Topics for Test 1 Matrix algebra Wiliamson Trotter 2 3 2 4 0 Matrix addition and scalar multiplication 0 Matrix multiplication o Diagonal matrices identity matrix 0 Matrix polynomials o Inverse matrix Determinants WilliamsonTrotter 25 o Explicit formulas for 2gtlt2 and 3gtlt3 matrices 0 Elementary row and column operations 0 Row and column expansions 0 Test for linear dependence Sample problems for Test 1 Problem 1 25 pts Let Fl be the plane in R3 passing through the points 200 110 and 730 2 Let Z be the line in R3 passing through the point 111 in the direction 2 22 i Find a parametric representation for the line Z ii Find a parametric representation for the plane Fl iii Find an equation for the plane Fl iv Find the point where the line Z intersects the plane Fl v Find the angle between the line Z and the plane Fl vi Find the distance from the origin to the plane Fl Problem 2 15 pts Let fx acos 2X bcosx c Find a b and c so that f0 0 f 0 2 and I M0 10 Sample problems for Test 1 0 72 4 1 2 3 2 0 Problem 3 20 pts Let A 1 0 71 1 1 0 0 1 Find the inverse matrix A l Problem 4 20 pts Evaluate the following determinants 072 41 27203 2320 quoti5 321 01 071139 390 17107339 1001 20071 Bonus Problem 5 15 pts Find the volume of the tetrahedron with vertices at the points a 10 0 b 010 c 001 and d 235 Problem 1 Let Fl be the plane in R3 passing through the points 200 110 and 302 Let E be the line in R3 passing through the point 111 in the direction 222 i Find a parametric representation for the line 6 Parametric representation t2 2 2 111 The line 6 passes through the origin t 12 Hence an equivalent representation is s222 Problem 1 Let Fl be the plane in R3 passing through the points 200 110 and 302 Let E be the line in R3 passing through the point 111 in the direction 222 ii Find a parametric representation for the plane H Since the plane Fl contains the points a 200 b 110 and c 302 the vectors b a 110 and c a 502 are parallel to H Clearly b a is not parallel to c a Hence we get a parametric representation t1b a t2c aa t1 1 10 t2 502 200 Problem 1 Let Fl be the plane in R3 passing through the points a 200 b 110 and c 302 Let E be the line in R3 passing through the point 111 in the direction 222 iii Find an equation for the plane H Vectors b 7 a 7110 and c 7 a 7502 are parallel to Fl gt their cross product p is orthogonal to Fl ljk 10710711 p 7110 1 7 HL k 7502 02 52 50 2i2j5k225 Apoint xXyz is in the planel39lifand only if pxia0 ltgt 2X722y520 ltgt 2x2y524 Problem 1 Let Fl be the plane in R3 passing through the points 200 110 and 302 Let E be the line in R3 passing through the point 111 in the direction 222 iv Find the point where the line 6 intersects the plane H Let x0 Xyz be the point of intersection Then x0 s222 for some 5 E R and also 2X 2y 52 4 225 225 525 4 ltgt s 29 Hence x0 49 49 49 Problem 1 Let Fl be the plane in R3 passing through the points 200 110 and 302 Let E be the line in R3 passing through the point 111 in the direction 222 v Find the angle between the line 6 and the plane H Let q denote the angle between vectors u 2 2 2 and p 225 Our angle is 11 l7T2 up 18 3 cosq lul lpl 12 33 11 1 7T 3 3 arccos arcsm 2 11 11 Problem 1 Let Fl be the plane in R3 passing through the points 200 110 and 302 Let E be the line in R3 passing through the point 111 in the direction 222 vi Find the distance from the origin to the plane H The equation of the plane Fl is 2X 2y 52 4 Hence the distance from a point X0y0Z0 to Fl equals 2X0 2y0 5Z0 i 2X0 2y0 5Z0 22 22 52 39 The distance from the origin to the plane is equal to Mm Bonus Problem 5 Find the volume of the tetrahedron with vertices a 100 b 010 c 001 and d 235 Vectors x b 7 a 7110 y c 7 a 7101 and z d 7 a 13 5 are represented by adjacent edges of the tetrahedron 1 It follows that the volume of the tetrahedron is glx y gtlt 71 1 0 xygtltz 71 0 1 71 g E 71 71 E 9 1 3 5 Thus the volume of the tetrahedron is 1 1 6xygtltz6915 Parallelepiped is a prism Volume area of the base x height Area of the base y gtlt z Volume x y x z Tetrahedron is a pyramid Volume area of the base x height Area of the base y gtlt z gt Volume x y x z Problem 2 Let fX acos2X bcosx C Find a b and C so that f0 0 f 0 2 and f 0 10 f X 743 cos 2X 7 bcosx f X 16a cos 2X bcosx gt f0 a b c f 0 74a 7 b f 0 16a b The coefficients a b c should satisfy a system abc0 abc0 a1 74a7b2 ltgt 74a7b2 ltgt b76 16ab10 12a12 Thus fX c052X 7 6cosx 5 0 2 4 1 i 2 3 2 0 1 Problem 3 Let A7 1 0 71 1 FIndA 1 0 0 1 First we merge the matrix A with the identity matrix into one 4gtlt8 matrix Ail i i IO OHMb 1 0 0 0 1 1 1 0 oowm i OOOH cor 0 l OOO Then we apply elementary row operations to this matrix until the left part becomes the identity matrix Interchange the lst row with the 4th row 1 0 0 1 0 0 0 1 2 3 2 0 0 1 0 0 1 0 1 1 0 0 1 0 0 2 4 1 1 0 0 0 Subtract 2 times the lst row from the 2nd row 1 0 0 1 0 0 0 1 0 3 2 2 0 1 0 2 1 0 1 1 0 0 1 0 0 2 4 1 1 0 0 0 Subtract the lst row from the 3rd row 1 0 0 1 0 0 0 1 0 3 2 2 0 1 0 2 0 0 1 0 0 0 1 1 0 2 4 1 1 0 0 0 Add the 4th row to the 2nd row 1 0 0 1 0 0 0 1 0 1 6 1 1 1 0 2 0 0 1 0 0 0 1 1 0 2 4 1 1 0 0 0 Add 2 times the 2nd row to the 4th row 1 0 0 1 0 0 0 1 0 1 6 1 1 1 0 2 0 0 1 0 0 0 1 1 0 0 16 1 3 2 0 4 Add 16 times the 3rd row to the 4th row 1 0 0 1 0 0 0 1 0 1 6 1 1 1 0 2 0 0 1 0 0 0 1 1 0 0 0 1 3 2 16 20 Multiply the 3rd and the 4th rows by 1 1 0 0 1 0 0 0 1 6 1 1 1 0 0 1 0 0 0 1 1 0 0 0 1 3 2 Add the 4th row to the 2nd row 1 0 0 1 0 0 0 1 0 1 6 0 2 1 16 18 0 0 1 0 0 0 1 1 0 0 0 1 3 2 16 20 Subtract the 4th row from the lst row 1 0 0 0 3 2 16 19 0 2 1 16 18 0 0 0 1 1 1 3 2 16 20 000 OCH 6 1 0 Subtract 6 times the 3rd row from the 2nd row 1 0 0 0 3 2 16 19 1 0 0 2 1 10 12 0 1 0 0 0 1 1 0 0 1 3 2 16 20 I i A4 000 Finally the left part of our 4gtlt8 matrix is transformed into the identity matrix Therefore the current right part is the inverse matrix of A Thus 71 0 2 4 1 3 2 16 19 A71 2 3 2 0 7 2 1 10 12 1 0 1 1 0 0 1 1 1 0 0 1 3 2 16 20 2 Problem 4i A Find detA l l MO 41 20 11 01 COCO In the solution of Problem 3 the matrix A has been transformed into the identity matrix using elementary row operations Those included one row exchange and two row multiplications each time by 1 gt det 12 detA gt detA det 1 5 3 2 1 Problem 4II Bi 1 1 0 3 Flnd detB 1 Expand the determinant by the 3rd column 2 203 223 5 32 1 7 21 1 3 1 10 3 201 2 00 1 Subtract 2 times the 2nd row from the lst row Subtract 2 times the 2nd row from the lst row 2 2 3 0 0 9 detB 2 1 1 3 2 1 1 3 2 0 1 2 0 1 Expand the determinant by the lst row 0 0 9 1 1 detB 2 1 1 3 292 0 2 0 1 Thus 1 1 detB 18 2 0 182 36

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