Chapter 3.1 Vectors

We represent p as the triple x0 y0 z0 the coordinate

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Unformatted text preview: through the z -axis at z0 . We represent P as the triple (x0 , y0 , z0 ). The coordinate axes in 3-dimensional space divide space into 8 octants determined by the eight possible sign combinations of the three coordinates. The octant consisting of points with all positive coordinates is called the first octant. Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Vectors in Space Denote Cartesian three space as R3 . A vector v in R3 is a directed line segment from an initial pointP (x0 , y0 , z0 ) to a terminal point Q (x1 , y1 , z1 ). z Similar to vectors in the plane, v is represented with an ordered triple v = v1 , v2 , v3 , where Q P v1 = x1 − x0 v2 = y1 − y0 v3 = z1 − z0 . The numbers v1 , v2 , and v3 are called the components of v. x y Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Vector Algebra by Components Theorem Suppose v = v1 , v2 , v3 and w = w1 , w2 , w3 , and let λ be a scalar. (a) v + w = v1 + w1 , v2 + w2 , v3 + w3 . (b) λv = λv1 , λv2 , λv3 . (c) v − w = v1 − w1 , v2 − w2 , v3 − w3 . Example 1 Find the sum of the vectors v = 7, −3, 2 and w = −5, 8, 4 . 2 1 If v = 6, −4, 3 , then find −5v and 2 v. 3 Find the components of v having initial point P (2, 4, −3) and terminal point Q (7, 5, 4). Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Translation of Axes Sometimes it is convenient to translate the coordinate axes to obtain a new coordinate system with axes parallel to the original system but with a new origin. If the translated origin has coordinates O (p , q , r ) in the original system, then a point P has coordinates P (x , y , z ) in the original system and coordinates P (x , y , z ) in the translated system. There is a nice relationship between the two coordinates given by constructing the position vector −→ − O P = x , y , z = x − p, y − q, z − r . Since the vector is the same regardless of the coordinate system, we must have: x = x − p, y = y − q, z =z −r Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space Example Example If an xyz -coordinate system is translated so that the origin is O (1, −3, 5), find P (2, 0, 4) in the new coordinate system. The Norm Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Properties of the Norm Theorem Suppose v and w are vectors, and λ is a scalar. (a) λv = |λ| v . (b) v + w ≤ v + w (the triangle inequality). Example 1 Find the norm of v = −5, 3, 2 . 2 Use vectors to find the distance between P (1, 3, 0) and Q (−3, −4, 6). 3 Verify the Triangle Inequality for the vectors v = 4, −3 and w = 5 12. Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Unit Vectors If a vector has length 1, then it is called a unit vector. Set i = 1, 0, 0 and j = 0, 1, 0 , and k = 0, 0, 1 . These unit vectors can be used to write any vector v = v1 , v2 , v3 in a unique way as a linear combination of the form: v = v1 i + v2 j + v3 k....
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