1. Given the vectors ū=(1,-2,-6) and v = (0,-3,4), a) Find u 6v. b) Find a unit vector in the opposite direction to ū. c) Find (ü.v)v. d) Find 11: e) Find the distance between ū and v. f) Are ū and y parallel, perpendicular, or neither? Explain. g) Verify the Triangle Inequality for ū and ū.
3. Consider the following vectors, where k is some real number. H-11 Lol 1-1 a. For what values of k are the vectors linearly independent? b. For what values of k are the vectors linearly dependent? c. What is the angle (in degrees) between u and v? 4. Here are two vectors in R". Let V = the span of {"v1r2} a. Find an orthogonal basis for V (the orthogonal complement of V). b. Find a vector that is neither...
How to solve all of this linear
Algebra
8. (24 points total) LetV be the vector space{P2, +, *}with standard function addition and scalar multiplication Define an Inner product: <p | q>= p(0)q[O) + p(1)q(1)+ p(2)q(2). Let B = {x,x,1} a. Explain why this inner product satisfies the positive property b. Explain how you know that B forms a basis c. State the conclusions of Cauchy-Schwartz and the Triangle inequalities in terms of this inner product d. Use Gram-Schmidt and...
Consider the points u(1, 1,-1), v (a, 2,-1) and w (1,2,-1) in R3, where a e R. There are two possible values of a for which u, v and w will form an isosceles triangle. a) Find one of these values. (b) Determine the angle between the equal sides of the triangle.
Consider the points u(1, 1,-1), v (a, 2,-1) and w (1,2,-1) in R3, where a e R. There are two possible values of a for which u, v...
1) Consider u = 2 -2), v 1 2 and w=3, where a is real number. -- a) Find the length of w. b) Find the distance between u and v. c) Find a unit vector in the direction of w. d) Find the real number a such that v and w are orthogonal. e) Find the angle 0 between u and v. remote proctor each individualsheet of paper front and
Linear Algebra
2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
Suppose that u and v are non-zero vectors in Rn. Verify that the two vectors u and v - (u.v/u.u)u are orthogonal. Then pick two specific vectors u and v in R2. Plot the three vectors, u, v and v - (u.v/u.u)u on the same graph. Explain the geometric significance of v - (u.v/u.u)u
In R. let V be the orthogonal complement of the vectors u and v, where u = (1,9, 3,61) and v= (4, 36, 13, 254) Find a basis B = {b1,b2} for V: b = 1 Now find five vectors in V such that no two of them are parallel e- LLL
7. Consider the vectors v = 3i - 5j, and u = -41 - 2j. Determine whether these vectors are orthogonal. If not orthogonal, determine the angle they form, to the nearest tenth of a degree.
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)