Let P(0,1,0), Q(2,1,3), R(1,-1,2). (la) Compute PQxPR. (1b) Find the equation of the plane through P,...
(1) Equation of a Plane Let P(1,1,-1), Q(1,2,0), R(-2,2,2). (la) Compute PQxPR. (1b) Find the equation of the plane through P, Q and R in the form ax+by+cz=d. (10) What is the angle formed by this plane and the xy-plane?
3. Let S be the plane through the points A(0,0,-1), B(0,2,0), C(3,0,0). (a) Find an equation for S of the form ax +by+cz = d. (b) Find the point on that is closest to the point P(2,2,3).
the plane PQ X PR 1. Find unit vector the perpendicular to P(1,1,1), Q(2,1,3), R(2, 2, 1).
-n ', S Let f(x,yZFz2_xy. Let v=<1,1,1>. Let point P=<2,1,3> a. Compute gradient of fx,y,z) b. If the contours are far apart, is the length of the gradient large or small? Answer: Explain! What MATLAB command is used to draw the gradient vectors? Answer: - c. Compute the directional derivative in the direction of v. d. Compute the equation of the tangent plane to f(x,y,z) at the point P. e. Use the chain rule to compute r if x t2,...
Let k ⊂ R 3 be the plane given by the equation ax+by+cz=d, can plane k written or express as a span of a set vectors? Justify your answer.
Consider the points: P (-1,0, -1), Q (0,1,1), and R(-1,-1,0). 1.) Compute PQ and PR. 2.) Using the vectors computed above, find the equation of the plane containing the points P, Q, and R. Write it in standard form. 3.) Find the angle between the plane you just computed, and the plane given by: 2+y+z=122 Leave your answer in the form of an inverse trigonometric function.
in the direction of the vector OR. Put your answer in the Given the points P 2.3.1). Q (-1.1.2) and R (1.1.0) a) (3 pts) find an equation for the line that passes through the point form r(t) = (x(t), y(t), z(t)). b) (4 pts) find a non-zero vector normal to both Po and QR c) (3 pts) find an equation for the plane containing the points P Q and R. Put your answer in the form ax+by+cz =d.
Find an equation of the tangent plane to the following parametric surface, r(u, v) = (u2 + 9) i + (v3 + 6u) j + (u + 2v) k , at the point (10, 5, −1). Write the equation in the form ax + by + cz + d = 0, where a, b, c, and d have no common factors. Then enter the values of a, b, c, and d (in that order) into the answer box below, separated...
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)
In this problem, let li be the line that passes through the points A(1,2, 4) and B(-1,3,8), and let l2 be the line with symmetric equations x +1 = 2y = 32 — 3. Parts (e) and (f) relate to the vector field F = (xy, xz, yz). (a) Show that the lines li and l2 intersect. (b) Let P be the plane that contains both lines li and lz. Find an equation for P. (c) Show that the points...