Chapter 3.1 Vectors

Chapter 3.1 Vectors

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Unformatted text preview: e Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Vector Algebra by Components Theorem Suppose v = v1 , v2 and w = w1 , w2 , and let λ be a scalar. (a) v + w = v1 + w1 , v2 + w2 . (b) λv = λv1 , λv2 . (c) v − w = v1 − w1 , v2 − w2 . Example 1 2 Find the sum of the vectors v = −5, 2 and w = 7, 1 using componentwise addition, then verify that the parallelogram law yields the same vector. 1 If v = −5, 2 , then find 3 v and −3v. Describe the new vectors. Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space Properties of Vector Algebra Theorem Suppose u, v, and w are vectors in the plane. Let λ and µ be scalars. (a) v + w = w + v. (b) (v + w) + u = v + (w + u). (c) v + 0 = v = 0 + v. (d) v − v = 0. (e) λ(µv) = (λµ)v. (f) λ(v + w) = λv + λw. (g) (λ + µ)v = λv + µv. (h) 1v = v. The Norm Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Cartesian Coordinates of a Point in Space We set up a Cartesian coordinate system in 3-dimensional space as follows. Fix a plane and set up a Cartesian coordinate system in this plane. Call it the xy -plane, with coordinate axes labeled x and y , respectively. There is a line perpendicular to the xy -plane passing through the origin. Call this line the z -axis. Define the positive direction on the z -axis as the direction your thumb points if you curl the fingers of your right hand from the positive x -axis toward the positive y -axis through the smallest angle. Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Cartesian Coordinates of a Point in Space We have three Cartesian planes: the xy -plane, the xz -plane, and the yz -plane. Any point P in space lies at the intersection of three planes: One parallel to the yz -plane and passing through the x -axis at x0 ; One parallel to the xz -plane and passing through the point y0 on the y -axis; One parallel to the xy -plane and passing...
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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.

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