5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu...
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
5. Let u = [0,1,1), v = (-5, -4,6.7), and P = (4.–5.6). In the following, when rounding numbers, round to 4 decimal places. (i) Find the parametric form of the equation of the plane P. containing P and with direction vectors u and v. (ii) Find the parametric form of the equation of one of the two planes that are parallel to P, and distance 1 away from P1.
5. Let u = [0,1,1), v= (-5, -4,6.7], and P = (4, -5,6). In the following, when rounding numbers, round to 4 decimal places. (i) Find the parametric form of the equation of the plane Pi containing P and with direction vectors u and v. (ii) Find the parametric form of the equation of one of the two planes that are parallel to P1 and distance 1 away from P1.
Let R2 have the Euclidean inner product. (a) Find wi, the orthogonal projection of u onto the line spanned by the vector v. (b) Find W2, the component of u orthogonal to the line spanned by the vector v, and confirm that this component is orthogonal to the line. u =(1,-1); v = (3,1) (a) wi = Click here to enter or edit your answer (0,0) Click here to enter or edit your answer (b) 2 = W2 orthogonal to...
Let u = (2,3), v = (-5, 6), and w = (9,0). (a) Draw these vectors in R2 y 10 10 10 5 -10 -5 5 10 -10 -5 10 -10 -5 10 -10 o y 10 -10 -5 10 -10 O X (b) Find scalars 1, and in such that w = 1,0 + 12v. (11.12) - -1,2
U is a 2 x 2 orthogonal matrix of determinant -1. Find 5 · [0, 1] · U if 5 · [1,0] · U = (-3,4]. 2. Let M = [[144, 18], [18, 171]]. Notice that 180 is an eigenvalue of M. Let U be an orthogonal matrix such that U-MU is diagonal, the first column of U has positive entries, and det(U) = 1. Find 145 · U.
Help would be greatly appreciated!! 1. Let S be the surface in R3 parametrized by the vector function ru, v)(,-v, v+ 2u) with domain D-{(u, u) : 0 u 1,0 u 2). This surface is a plane segment shaped like a parallelogram, and its boundary aS (with positive orientation) is made up of four line segments. Compute the line integral fos F -dr where F(z, y, z) = 〈エ2018 + y, 2r, r2-Ins). Hint: use Stokes' theorem to transform this...
17 Find the orthogonal complement of the following. a. U = sp({(3,-1,2)}) in R3. b. V=({(1,3,0), (0,2,1))) in R3. Do this both algebraically and geometrically. Compare with part a. c. W=sp({1+x}) in 81 (-1,1]).
2. Let v= [6, 1, 2], w = [5,0, 3), and P= (9, -7,31). (i) Find a vector u orthogonal to both v and w. (ii) Let L be the line in R3 that passes through the point P and is perpendicular to both of the vectors v and w. Find an equation for the line L in vector form. (iii) Find parametric equations for the line L.
2. Consider R with the weighted inner product = [wn, u, tva, teal"). [ruh, t', talT and w Find the orthogonal projection of w = [1, 2,-1,2]T onto the span of ui-|1,-1, 2, 5]T and u2 [2,1,0,-]. Make sure you are working with an orthonormal basis for u span(u, u2 before you use the usual projection formula. 2. Consider R with the weighted inner product = [wn, u, tva, teal"). [ruh, t', talT and w Find the orthogonal projection of...