please provide clear drawings for each 5. Let T9,b) (x, y) = (x +a, y +b)...
(0. 0), (0.2), and e2an model for uclidean geometry. Find the images of the points (0, 0), (0,2), and (2, 1) under the following isometries. ssume the Cartesian mod 1. A (a) The reflection across the line with equation x y. (b) The reflection across the line x y 5. (c) The half-turn about (-1, 1). (d) The rotation through 45e in the counterclockwise direction about the point (2,2). (e) The glide reflection GAB, where A (-2.0) and B (0.1)...
Let T: R2 R2 be a reflection in the line y = -x. Find the image of each vector. (-3,5) (b) (7.-1) (c) (-a, 0) (d) (0, b) (e) (e, -d) (f) (9)
let F(x,y) = 3x^2y^2i+2x^3yj and c be the path consisting of line segments from(1,2) to (-1,3), from (-1,3) to (-1,1), and from (-1,1) to (2,1). evaluate the line integral of F along c. Let F(x, y) = 3x²y2 i + 2x’yj and C be the path consisting of line segments from (1, 2) to (-1,3), from (-1, 3) to (-1, 1), and from (-1, 1) to (2, 1). Evaluate the line integral of F along C.
(c) [1 point] Let R : E3 → E3 be the rotation in E3 with axis in the direction of the vector ã=(-1,2, -2) and angle 0 = . If pe E3 denotes the point (0,0,1) then ... R(p) = (d) [1 point] Let R: E2 → Eº be a reflection through a line l that fixes the origin and sends (1,1) to some point on the line y = x. Can you determine the line l? If so, give...
5)Evaluate f (x2-y)dr + (y2 +x)dy along a) a straight line from (0.1) to (1,2), b) straight lines from (0,1) to (1,1) and then from (1,1) to (1,2), c) the parabola z t,y 1
(1 point) Let F = xi+ (x + y) 3+ (x – y+z) k. Let the line l be x = 4t – 3, y = — (5 + 4t), z = 2 + 4t. = (20, Yo, zo) where F is parallel to l. (a) Find a point P P= Find a point Q = (x1, Yı, z1) at which F and I are perpendicular. Q - Give an equation for the set of all points at which F...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
Question 2 (5%). Let L be a line in 3D such that it is perpendicular to the plane X - Y - z = 1. Choose the correct answer for vectors which are perpendicular to L. (A) All normal vectors of the plane X – Y – z = 10. (B) All normal vectors of the plane x + 2y + 3z = 3. (C) All vectors parallel to the line (-1,-1,1) ++(-1,-1,1). (D) All vectors parallel to the line...
1. Let L: R2-R2 be defined by L(x.y) (x +2y, 2x - y). Let S be the natural basis of R2 and let T = {(-1,2), (2,0)) be another basis for R2 . Find the matrix representing L with respect to a) S b) S and1T c) T and S d) T e) Find the transition matrix Ps- from T basis to S basis. f) Find the transition matrix Qre-s from S-basis to T-basis. g) Verify Q is inverse of...
Let X and Y have the joint pmf defined by (х, у) (1,2) (0,0) (0,1) (0,2) (1,1) (2,2) 2/12 1/12 3/12 1/12 1/12 4/12 Pxy (x, y) Find py (x) and p, (y) а. b. Are X and Y independent? Support your answer. Find x,y,, and o, С. d. Find Px.Y