1. Let u - (1,1,2), v = (1,2,1), and w (2,1,1) in R. and consider • the parallelogram B = {s(3v) + t-w) 0 <s,t<1, s,te R} spanned/formed by the vectors (3v) and (-w), and • the parallelepiped P = {ru + s(3v) + (-w) 0 <T,,t<1, r, s, t€ R} [10] spanned formed by vectors u. (3v). and (-w) We take the parallelogram B as a base of P. (a) Does the ordered vector triple (v xw, 3v, -w),...
(1) Let u = (-1,2) and v = (3, 1). (a) (5] Find graphically the vector w = (2u - v). (b) (5] Find algebraically the vector z=3u - 2 (2) (a) [5] Write u ='(1, -5, -1) as a linear combination of v1 = (1,2,0), v2 = (0,1,-1), V3 = (2,1,1). (b) (5] Are the 4 vectors u, V1, V2, V3 linearly independent? Explain your answer. (C) (5) Are the 2 vectors V, V3 linearly independent? Explain your answer....
(1 point) Are the following statements true or false? ? 1. u? v – vſ u = 0. ? 2. If x is orthogonal to every vector in a subspace W , then x is in Wt. ? 3. For any scalar c, ||cv|| = c||v. ? 4. For an m x n matrix A, vectors in the null space of A are orthogonal to vectors in the row space of A. ? 5. If u and v are nonzero...
Let u = [1, 3, -2], v = [-1, 1, 1], w = [5, 1, 4]. a) Check if the system of vectors {u,v,w} is an orthogonal or othonormal basis of E3. b) Find the coordinates of the vector [1,0,1] in this basis.
3 - 2 Let u= Note that {u, v, w} is an orthogonal set of vectors and w - -3 4 9 be a vector in subspace W, where W = Span{u, v, w}. Let y= 11 -27 Write y as a linear combination of u, v, and uw, i.e. y = ciu + cqũ + c3W. Answer: y=
Let w be a subspace of R", and let wt be the set of all vectors orthogonal to W. Show that wt is a subspace of R" using the following steps. a. Take z in wt, and let u represent any element of W. Then zu u = 0. Take any scalar c and show that cz is orthogonal to u. (Since u was an arbitrary element of W, this will show that cz is in wt.) b. Take z,...
u and v are perpendicular. Find the triple scalar product of u, v and w=-3⋅u×v+2⋅u+6⋅v if |u|=6, |v|=8.
Q1. Given the points A: (0,0,2), B: (3,0,2), C: (1,2,1), and D: (2, 1,4 a) Find the cross product v - AB x AC. b) Find the equation of the plane P containing the triangle with vertices A, B, and C c) Find u the unit normal vector to P with direction v d) Find the component of AD over u and the angle between AD and u, then calculate the volume of the parallelepiped with edges AB, AC, AD...
QUESTION 10 Find the triple scalar product (u x v). w of the vectors u = 2i - 4j, v= -4i - 6j + 4k, w=9i - 7j+3k ОА 74 OB 140 OC-140 OD 458 ОЕ 74
1- Two vectors are given as u = 2 – 5j and v=-{+3j. a- Find the vector 2u +3v (by calculation, not by drawing). (4 pts) b- Find the magnitudes luand il of the two vectors. (4 pts) c- Calculate the scalar product u•v. (5 pts) d- Find the angle between the vectors u and v. (6 pts) - Calculate the vector product uxv. (6 pts)