Download Vector Operations and Component Vectors in 2D and 3D and more Exams Linear Algebra in PDF only on Docsity! Math 311 Lecture 8 Exam 1 Wednesday, covers Lectures 1-8. DEFINITION. A vector in the plane or 3-space is a directed line segment PQ¯ ¯ ̄directed from a tail P to a head Q. Two vectors are equal iff one can be translated to the other iff they have the same length (also called magnitude) and direction. The components of the vector are the (head point coordinates) (tail point coordinates). These are also the coordinates of its head when its tail is at the origin. The components form 2[1 or 3[1 column matrices which are also called vectors or component vectors. (3,4) (0,0) (5,-2) (8,2) head tail These vectors are all equal. The three directed line segments above are all regarded as being the same vector and are identified with the 2[1 column matrix Its magnitude is = 5. 3 4 . 32 + 42 DEFINITION. For any vectors u and v and any scalar a, av = the result of increasing the length of v by a times. If a < 0, the direction of v is reversed. u+v = the third side of the triangle formed by putting the tail of v on the head of u = a diagonal of the parallelogram formed by putting the tails of u and v together. uv = the vector from the tail of u to the tail of v when the heads of u and v are put together. v 2v -v v v u u u+v u+v THEOREM. The components of a sum or scalar product when added above are the sum and product of the components when regarded as 2[1 or 3[1 matrices. Hence the same associativity, commutativity and distributivity rules hold. In 3-space, we shall use a right-handed coordinate system where the x-axis comes forward out the plane, the y-axis goes to the right and the z-axis goes up. y-axis z-axis x-axis (x,y,z) CWhat is the component vector for the vector PQ¯ ¯ ̄where P = (0, 1) and Q = (1, 0). CWhat is the component vector for the vector PQ¯ ¯ ̄where P = (3, 4, 5) and Q = (1, 2, 3). CFor the vector , 1 2 (a) if the tail is at (3,3), where is the head? (b) if the head is at (3,3) where is the tail? CSuppose v = (0,1)(3,3)¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ and u = (1,1)(2,2)¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯. Find the components of each vector: v = u = v + u = 2v u = CFind a and b such that a = b = a 1 −1 + b 2 2 = 0 1 Hw 6 Answers Page 62. 2ò (b) (c) 6(2). 1 0 0 0 −2 0 0 0 1 1 0 0 0 1 3 0 0 1 0 0 1 0 1 0 1 0 0 1 −1 0 3 2 1 2 −3 2 −1 0 1 8(8). (a) No inverse (b) 3 1 2 1 -1 0 2 1 2 1 -2 1 1 2 2 -3/2 5/2 -1/2 (c) 1 2 3 -1 3/2 1/2 1 1 2 1 -3/2 1/2 1 1 0 0 1/2 -1/2 (d) 1 2 3 3/4 -3/2 1/4 0 1 2 1/2 0 -1/2 1 0 3 -1/4 1/2 1/4 12(2). Many possible answers. One is A = 1 0 0 0 1 0 1 0 1 1 2 0 0 1 0 0 0 1 1 0 0 0 1 0 0 −2 1 1 0 0 0 1 0 0 0 4 1 0 0 0 1 2 0 0 1 1 0 −1 0 1 0 0 0 1
