3. If ū= 4.2,1 and ū= -2.2.1), find a vector in R3 that is orthogonal to...
(11 Let u Show that B } is an orthogonal basis of R3. (b) Convert B into an orthonormal basis C of R3 by normalizing ü, ū and w. Show your work. Find the change of coordinates matrices Psee and Pee-swhere C is the or- thonormal basis of R3 you found in (b) and S is the standard basis of R3. Justify your answers. Suppose now that ü, ū and w are eigenvectors of a 3 x 3 matrix A...
1. Let ū= (2,4,-1), v = (3.-3,-1) (a) Compute: x ū (b) Compute: ü x 7 (c) Is the cross product commutative? If not, what is it instead? 2. Let A = (7, -11,3), B = (1,9, -3), C = (-6,3, -2), D= (0,-8, 12), E = (1, -13,2) (a) Give the vector equation of a line passing through the points A, B. (b) Find the equation of the plane containing the points C,D,E. (c) Find the point of intersection...
Given the following vectors: ū= 3 ū= W = > (a) Find the projection of ū onto ū. BOX YOUR ANSWER. (b) Find the projection matrix of the projection in part (a). BOX YOUR ANSWER. (c) Find the projection of ū onto the subspace V of R3 spanned by ✓ and W. (You may use MATLAB for matrix multiplication in this part, but you must provide the expressions in terms of matrices.) BOX YOUR ANSWER. (d) Find the distance from...
Given in space the points A(4,7,1), B(2,1,3), and c(0,-1,2) The vectors ū = AB , and ✓ = AC a. (9%) Find ū. v , ū x ū , proj, u b. (3%) Find the area of triangle ABC. c. (3 %) Find the parametric equation of line (AB). d. (3 %) Find the distance from point C to the line (AB). e. (3 %) Find the equation of the plane (ABC). A relatively easy way of getting into international...
2. (a) Show that is an orthogonal basis for R3. (b) Find a non-zero vector v in the orthogonal complement of the space 0 Span 2,2 Do not simply compute the cross product. (c) Let A be a 5 × 2 rnatrix with linearly independent columns. Using the rank-nullity theorem applied to AT, and any other results from the course, find the dinension of Col(A) 2. (a) Show that is an orthogonal basis for R3. (b) Find a non-zero vector...
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]).
Part c and d Question 5 (30 marks) Let A1, B(3,-5,0) and C(-1,4,1) be three points in R. Use vector method(s) to solve each of the following. R-8 (a) Find the unit vector u in the direction of AB-3AC. (b) Calculate the smaller angle betwen AB and AC. Correct the answer to ONE decimal place. (c) Find the shortest distance between B and the line passing through A and C. Correct the answer to ONE decimal place. (Hint: Consider the...
Let E be the plane in R3 spanned by the orthogonal vectors v1=(121)and v2=(−11−1) The reflection across E is the linear transformation R:R3→R3 defined by the formula R(x) = 2 projE(x)−x (a) Compute R(x) for x=(1260) (b) Find the eigenspace of R corresponding to the eigenvalue 1. That is, find the set of all vectors x for which R(x) =x. Justify your answer.
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 2 Consider the vector space R3 (2.1) Show that (12) ((a, b, c), (x, v, z))-at +by +(b+ c)(y + z) is an inner product on R3 (2.2) Apply the Gram-Schmıdt process to the following subset of R3 (12) to find an orthogonal basis wth respect to the inner product defilned in question 2.1 for the span of this subset (2.3) Fınd all vectors (a, b, c) E R3 whuch are orthogonal to (1,0, 1) wnth respect to the...