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MATH 2120 W13 Review 1 1 1. Find the three angles of the triangle △ABC with vertices A(1, 0), B(3, 6), C(−1, 4). Ans. 45◦ , 45◦ , 90◦ . 2. For what values of b are the vectors (−6, b, 2) and (b, b2 , b) orthogonal? 3. Find a unit vector that is orthogonal to both (1, 1, 0) and (1, 0, 1). Ans. √1 (1, −1, −1). 3 4. Find the acute angle between the lines 2x − y = 3 and 3x + y = 7. Ans. 45◦ . 5. Find the vector projection of v = (2, −1, 4) onto u = (0, 1, 12 ). Ans. 1 (2, −1, 4). 21 6. Find the distance from the point (−2, 3) to the line 3x − 4y + 5 = 0. Ans. 7. 13 . 5 a) Find the angle between the planes x + y + z = 1 and x − 2y + 3z = 1. b) Find the equation of the line of intersection of the planes. x 1 5 Ans. a) 72◦ b) y = 0 + t −2 . z 0 −3 8. Find the √ distance between the parallel planes 10x + 2y − 2z = 5 and 5x + y − z = 1. Ans. 63 . 9. Find the equation of the plane through the point (6, 3, 2) and perpendicular to the vector v = (−2, 1, 5). Ans. −2x + y + 5z = 1. 10. Find the equation of the plane that passes through the point (−1, 2, 1) and contains the line of intersection of the planes x + y − z = 2 and 2x − y + 3z = 1. Ans. x − 2y + 4z = −1. 11. Find the point at which the line x = 3 − t, y = 2 + t, z = 5t intersects the plane x − y + 2z = 9. Ans. (2, 3, 5). 12. Prove that u · v = 14 ∥u + v∥2 − 14 ∥u − v∥2 for all vectors u, v in Rn . 13. Prove that u + v and u − v are orthogonal in Rn if and only if ∥u∥ = ∥v∥. 14. Solve the system of equations using either Gaussian or Gauss-Jordan elimination. w + x + 2y + z w−x−y+z x+y w+x+z Ans. No solution. = = = = 1 0 −1 2 MATH 2120 W13 Review 1 2 15. For what values of k, if any, will the sytems have a) no solution, b) a unique solution, and c) infinitely many solutions? x + y + kz = 1 x + ky + z = 1 kx + y + z = −2 16. Determine by inspection(without reducing the matrix) whether the linear system with the augmented matrix below has a unique solution, infinitely many solutions, or no solution. 1 2 3 4 5 6 7 7 6 5 4 3 2 1 8 8 8 8 8 8 8 Ans. No solution. 1 1 1 3 0 , 1 , and 1 . 17. Show that R is spanned by the vectors 0 0 1 3 0 4 geometrically. 0 and 18. Describe the span of the vectors 1 0 Ans. Line in R3 through the points (0, 0, 0) and (3, 4, 1). 0 2 2 19. Determine if the vectors u = 1 , v = 1 , and w = 0 are linearly 2 3 1 independent or not. If the vectors are linearly dependent, find a dependence relation among the vectors. Ans. Lin. dependent, −u + v = w. 20. Prove that two vectors in Rn are linearly dependent if and only if one is a scalar multiple of the other. ( ) ( ) 1 2 −1 0 21. Given A = and B = , solve the equation 2(A − B + 2X) = 3 4 1 1 3(X − B) for X. ( ) ( ) 1 2 a b 22. If A = and B = , find conditions on a, b, c, d such that AB = BA. 3 4 c d Ans. 3b = 2c, a = d − c. 23. Show that for m × n matrices A and B, (A + B)T = AT + B T . 24. Show that for any matrix A, the matrix AAT is symmetric.