University:
Seattle Pacific University
Course:
MAT 1236 | Calculus III
Academic year:

2024
Views:

340

Pages:

1
Author:

Brandon Owen

Workshop #2 1. Find the center and the radius of the sphere by writing it in the form (x − h)2 + (y − k)2 + (z − l)2 = r2 , and sketch the sphere. (a) x2 + y 2 + z 2 − 4x + 2y − 6z = 130 (b) 2x2 + 2y 2 + 2z 2 = 22 − 20x 2. Write equations or inequalities R3 that describe the set of points. Sketch a picture. (a) The solid rectangular region (a box!) in the first octant bounded by the coordinate planes and the planes x = 6, y = 1, z = 3. (b) The outside of the sphere (no boundary) that has center (0, 1, −4) and is tangent to the xy-plane. (c) The inside of the sphere with center at (−2, 5, 4) and passing through the point (0, 1, 0). What changes if we include the boundary? (d) The sphere, and its inside, that has a diameter connecting the points (3, 5, −3) and (7, 3, 1). 3. The vector ~v = h2, 5i is in R3 and is the position vector for P (2, 5). (a) Sketch this position vector starting at the origin. (b) Sketch ~v when it starts at the point (4, 1), and also when it starts at the point (−6, −2). In each case, where does it end? (c) How many different places can this vector be drawn? 4. A vector ~v in R3 has initial point (−2, 5, 4), and terminal point (3, 5, −3). (a) Find ~v in component form, and also in ~i,~j, ~k-form. (b) Find two unit vectors parallel to ~v. (c) Find a vector of length 4 that is parallel to ~v. ~ = h−4, 1, −2i. Compute the following. 5. We have two vectors in R3 , ~v = h1, 2, 3i, w (a) ~v − 3~ w (b) |~v|~ w