Question 2 (5%). Let L be a line in 3D such that it is perpendicular to...
a) Let L be the line through (2,-1,1) and (3,2,2). Parameterize L. Find the point Q where L intersects the xy-plane. b) Find the angle that the line through (0,-1,1) and (√3,1,4) makes with a normal vector to the xy-plane. c) Find the distance from the point (3,1,-2) to the plane x-2y+z=4. d) Find a Cartesian equation for the plane containing (1,1,2), (2,1,1) and (1,2,1)
(I point) Let F=21+(z + y) j + (z _ y + z) k. (1+4t). y = 4 + 2t, z = _ (1+t). Let the line l be x =- (a) Find a point P-(zo, 30, zo) where F is parallel to 1. Find a point Q (which F and I are perpendicular. Q= and l are perpendicular Give an equation for the set of all points at which F and l are perpendicular. equation:
(I point) Let F=21+(z...
Question 1 10 pts 1) Write a full sentence to answer each question. a) What information do you need in order to find an equation for a plane? b) Is 3x – 2y + 4z — 7 an equation for a plane? Explain. c) What information do you need in order to find parametric equations for a line? d) Are x = -1+ 3t, y= 2 + 4t, z= -3 + 7t, parametric equations for a line? Explain 2) Decide...
Please provide clear handwritings for answers and specific step
by step explanations of questions 3 and 4. Thank you.
3. Are the plane 6z 3y - 4z-12 and line L 2, y 32t, z2-2t parallel? If so, find the distance between them. If they are not parallel, but are intersecting (at a single point), find the point of intersection. If they are none of the above, draw a cat. 4. The line r(t) = 〈1, 1,1〉 +t(1,3,-1) and the plane...
Find the equation of the line that passes through the point (2,3,4) and is perpendicular to the plane 2x-y + 3z = 4 a. x=4+2t , y=2-t, z=7-3t b. x=2+2t , y=3-t, z=4+t c. x=2-2t , y=-3+t, z=4-3t d. x=-2+4t , y=5-2t, z=-2+6t e. another solution
Please, include the explanations to your solution!
Begin with the paraboloid z #x2 + y2, for 0 s z s 64, and slice it with the plane y-0. Let S be the surface that remains for y 20 (including the planar surface in the xz-plane) (see figure). Let C be the semicircle and line segment that bound the cap of S in the plane z 64 with counterclockwise orientation. Let F (4z 3y,4x+3z,4y+3x. Complete parts (a) through (c) below 64...
Additional Problems: (HINT: It suffices to consider Just what happens (DX c A. Show by example that (a x b xc* a with i, j and k:) B. Find a vector which is perpendicular to every vector parallel to the plane z+y 0. C. Find the line which is the intersection of the planes x + y 0 and 3y-z = 0. D. Explain why the vectors in the following form describe a plane (where both t and s are...
Please write neatly!
22. Let S denote the plane 2x +y+ 3z = 6 in the first octant with the upward normal, and C denote its triangular boundary. Use Stokes' Theorem to evaluate the line integral F dr where F = <2z - x, x +y +z, 2y-x>.
22. Let S denote the plane 2x +y+ 3z = 6 in the first octant with the upward normal, and C denote its triangular boundary. Use Stokes' Theorem to evaluate the line...
56. Let Li and L2 be the lines whose parametric equations are L]: x = 41, y = 1 -21, z = 2 + 21 L2: x = 1+1, y = 1-1, Z=-1+ 41 (a) Show that Li and L2 intersect at the point (2,0, 3). (b) Find, to the nearest degree, the acute angle between L and L2 at their intersection. c) Find parametric equations for the line that is perpen- dicular to L, and L2 and passes through...
Question 10: [8pt total] Consider the line in 3D space (2,5, 2) +t(-6, 4, 2). 10)a) [4pt] Determine vanishing point of the line when centrally projected onto the plane z = 2: Vanishing point: 10)a)ii) (4pt] Graph the central projection of the line onto the plane z = 2. y 6 2 2 8 -6 -4 -2 0 2 8 -2 61