In Exercises 5-8, determine cos where is the angle between u and v. --B] -[3]. - [3] 2 5. u= 6. u= 7. u= 8. u 2i - 3j + k, 23. u= 24. u = - [B] [³] - [3] V= ---- 2 V = 19. P = (2.5). Q = (6.2) 20. P = (7.6), Q = (4.1) 21. P = (-4, 2), Q = (6,2) 22. P = (-2,4). Q = (4.2) 25. u= In Exercises 23-26, find u, and u₂ such that u₁ = projqu. u, and u₂ are orthogonal, and u = u₁ + u₂. q= --[3] -[9] -[] -[3] 6 q= 6 -- 4 -2 v=i-2j+3k q= In Exercises 13-18, there are at most two three- dimensional vectors u that satisfy the given conditions. Determine these vector(s) u. u 13. u i=1, u.j= 3, 14. u i=0, u j = 0, 15. u i= 3, u k= 4, 16. u.i= 12, u 17. u (i+j) = 2, 18. u (i+j) = 2, 36. u= 2 2 37. u = 0 - In Exercises 19-22, u = OP, v = OQ and w = projqu. Find the point R such that w OR. Graph u, v, and w 2.3 The Dot Product and the Cross Product 38. u = i + j. 39. u = i- 2k, k= 3, u u In Exercises 36-39, find a vector w such that uw = 0 and v. w = 0. V = 11 A -(1.0.01 - V= v=i+k k= 4 k= 4 ||u|| = 5 u (j+ k) = 4, (j+k) = 3, v=j+3k D ||u|| = 13 40. A (-1, 1, 2), B=(2, 1, -1), C = (0, -2,4) (01.0) u. k = 1 u.k=2 In Exercises 40-41, find a vector w that is perpendicular to the plane containing the given points A, B, and C. 147 (2 3 1)

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.