3. (15 pts) Find the equation of the plane through the point A=(0,1,0)'and perpendicular to a...
Find an equation of the plane. The plane through the point (3, 0, 1) and perpendicular to the line x = 6t, y = 2 − t, z = 9 + 4t
4. (4 pts) Consider the surface z=x2y+y3.(a) Find the normal direction of the tangent plane to the surface through (1,1,2).(b) Find the equation of the tangent plane in (a).(c) Determine the value a so that the vector−→v=−−→i+ 2−→j+a−→k is parallel to the tangent plane in (a).(d) Find the equation of the tangent line to the level curve of the surface through (1,1). 4. (4 pts) Consider the surface z = z2y + y). (a) Find the normal direction of the...
(5) Equations for Planes. (a) Find an equation of the plane passing through (1,2,3) that is parallel to the plane r -y + 2z = 5. (b) Find an equation of the plane passing through the point (0,1,0) and containing the line r = (-t, 2t, 4t).
Find the equation of the line passing through the point (1,2,4) and perpendicular to the plane x−y + z = 3
1. Find the vector equation of the line (a) through the point (1, 3) with gradient 2, (b) through the points (3,-5) and (-2, 4), (c) * through the point (2,-1) and parallel to the line r. (41 – 3j) – 2 = 0, (d) through the point (-3,6) and perpendicular to the line 3x - 5y = 7
Find the equation of the line passing through the point (1,2,3), and perpendicular to the plane x + y + z = 6.
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)
1. (15 points) (a) (5 points) Find the equation of the plane a that contains points A(1,5,4) B(1,0, 1) and C(4, 0,5) (b) (5 points) Find the distance from the point D(2, 1,7) to this plane (c) (5 points) If plane 3 has equation y -3z+2x = 5, find a unit vector that is parallel to the intersection of a and B.
10. Write an equation for the plane containig the points (-7,2,1). (9.0,-2) and (-5, -1,2). Is this plane parallel, perpendicular or neither to the plane 2x - 3y + 2 = 5? 11. Consider the line that passes through the point (6, -5,2) and that is parallel to the vector (-1, 1, 3). (a) Find symmetric equations for this line (b) Find the point at which this line passes through the yz-plane.
(1) (a) Find the equation of the line, Li, which passes through the points A : (4,y,z) = (0, -5, -3) and B : (x, y, z)=(3, 1,0). (b) Find the equation of the line, Ly, which passes through the points C:(x, y, z)=(-1, -3,2) and D: (x,y,z) = (4,3,6). (c) Show that L and Ly are not parallel lines. (d) Write the parametric equations for L, and L2, and then show that the lines Li and L2 do not...