Would appreciate some help with these, thank you

(Section 4.1) 20 points 1. a. Let u = (1, 2, -3, 5, 0), v = (0, 4, -1, 1, 2), and w = (7, 1, -4, -2, 3). Find the components of i. ut w ii. 3(2u - v) ini. (3u - 5) - (2u+ 4w) iv . - ( 20 - 50 + 21 ) + 8 b. Find scalars C1, C2, and c3 for which the equation is satisfied, C1(-1, 0, 2) + c2(2, 2, -2) + c3(1, -2, 1) = (-6, 12, 4) (Section 4.2) 20 points 2. a. Find the Euclidean distance between u and v and the cosine of the angle between those vectors. State whether that angle is acute, obtuse, or 90. i. u = (1, 2, -3, 0), 0 = (5, 1, 2, -2) ii. u = (0, 1, 1, 1, 2), 0 = (2, 1,0, -1,3) b. Verify that the Cauchy-Schwarz inequality holds. i. u = (4, 1, 1), 7 = (1, 2, 3) ii. u = (1, 2, 1, 2, 3), 0 = (0, 1, 1, 5, -2) (Section 4.3) 20 points 3. a. Find the distance between the given parallel planes 2x - y + z = 1; 2x- y + z= -1. b. i. Show that U = (a, b) and w = (-b, a) are orthogonal vectors. ii. Use the result in part i. to find two vectors that are orthogonal to U = (2, -3). iii. Find two unit vectors that are orthogonal to v = (-3, 4).(Section 4.4) 20 points 4. a. Find the general solution to the linear system and confirm that the row vectors of the coefficient matrix are orthogonal to the solution vectors. 21 + 3x2 - 423 = 0 x1+ 2 2 + 323 = 0 b. i. Find a homogeneous linear system of two equations in three unknowns whose solution space consists of those vectors in IR3 that are orthogonal to a = (-3, 2, -1) and b = (0, -2, -2). ii. What kind of geometric object is the solution space? iii. Find a general solution of the system obtained in part i., and confirm that Theorem 3.4.3 of the textbook holds. (Section 4.5) 20 points 5. a. Find the area of the triangle in 3-space that has vertices P(1, -1, 2), Q(0, 3, 4) and R(6, 1, 8). b. Given u = (-1, 2, 4), v = (3, 4, -2), w = (-1, 2, 5), compute the scalar triple product u . (v x w) c. Given u . (3 x w) = 3, find the following, i. v. (ux w ) ii. (ux W) . J iii. U. (w x w)