8. 4 marks] (a) Suppose that a is the reflection across the line z y +2-0 and β is the reflection...
(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)...
QUESTION 1 (15 MARKS) a) Given the line Lj: I = 2 - 2t, y = 5 + 2t, z=t-1 and 1 1 - 2 L2 : =y-3 = 2 4 i. Check whether the lines Lị and L2 parallel, intersect or skewed? (5 marks) ii. Find the shortest distance from the point (1, 2, -1) to the line Li- (3 marks) b) Given two planes 71 : 20 - 4y +z = 5 and T2 : 7x + y...
1) Show that two lines are skew x+1 y+2 z+3 4:x=y=z and L: +7=5 2) Find the general equation of the plane containing the point P (1,2,3 ) and L, . 3) Find the point Q-the point of intersection the plane found in 2) and the line L. 4) Find the distance from the point (1,-1,2) to the line Lą.
Consider the following geometrical objects in R3: li: x=2 -21, y = -1, z = 2, ER 12: (x, y, z) = (0,5, -1) + (2, -1, 1), 1ER II : 3x + 4y - 2z = 1 II2 : 3x + 3y + z = -1 (a) Find the intersection point of , and 2. (b) Find the line 63 containing A(2,3,4) and is parallel to both II, and II. (c) () Determine whether II, and 2 are intersecting,...
Find the equation for a plane containing 3 points: A(2, 2,1) in the form: ax+by+cz+d = 0 C(0, -2,1). Put the plane equation B(3,1, 0) х — 3 z+2 = y+5 = 2 L: Find the intersection point between 2 lines whose symmetric equations are: 4 х-2 L, : у-2 = z-3 -3 Find the parametric equation for a line that is going through point A(2,4,6) and perpendicular to the plane 5х-3у+2z-4%3D0. Name: x-3y4z 10 Find the distance between 2...
Find the point of intersection of plane 4x+5y-52-4=0 and the following line: (x-4)/5 = (y+3)/3 = z/3 If they have a point of intersection, enter the x-value of point in the following box. If the line is on the plane, enter ON in the box. If the line is not on the plane, and they are parallel, enter P in the box.
+ y +z- 0 y-1 5. Solve: ANS: -(2) -(0) 6. Suppose: W Calculate and H -5W+4H ANS: 7. Write the system in #5 as a MATRIX EQUATION and a VECTOR EQUATION Matrix equation: Vector Equation: Find all h so that the system with the augmented matrix is INCONSISTENT 8. 1-1 7 a. h=7 b. h-7 c. h0 d. h 0 is: 9. The SPAN of the vector d. none of these c. the point: (0, 0) b. the line...
Consider the following. z = x2
+ y2, z = 36 − y, (6, -1, 37) (a) Find symmetric equations of the
tangent line to the curve of intersection of the surfaces at the
given point. x − 6 12 = y + 1 −2 = z − 37 −1 x − 6 1 = y + 1 12 = z
− 37 −12 x − 6 = y + 1 = z − 37 x − 6 12 =...
6 5.) (10 points) (a) Suppose TỈ : R2 → R2 s a reflection in the line y = z. what is A, the standard matrix of T1? (b) Suppose T : R2R is the counterclockwise rotation by 8 J. What is A2, the standard matrix of T2? (c) What is the image of (2,-4) under T- Ti o T2?
6 5.) (10 points) (a) Suppose TỈ : R2 → R2 s a reflection in the line y = z....
(a) Find symmetric equations for the line that passes through the point (4, -2, 6) and is parallel to the vector (-1, 3, -4) x+ 4-Y+ 2 3 z-6 -4 -(x +4) 3(y 2)-4(z +6). y+2 z-6 3 -(x-4) 3(y +2) -4(z- 6). o4-2-116 = Y - 2-z+6 3 (b) Find the points in which the required line in part (a) intersects the coordinate planes. 5 ,5,0 x ) point of intersection with xy-plane 10 7 point of intersection with...