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Chapter 3.1 Vectors

# Chapter 3.1 Vectors - Outline Vector Quantities Combining...

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Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Geometric Vectors Math 2270 K. J. Platt, Ph.D. Department of Mathematics Snow College Spring 2014

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Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Outline 1 Vector Quantities 2 Combining Vectors 3 Vectors in the Plane Vectors in the Plane Vector Algebra 4 Vectors in Space Cartesian Coordinates in Space Vectors in Space Translation of Axes 5 The Norm
Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Scalars and Vectors A scalar quantity is a quantity that can by modeled with a real number. Length Mass Temperature There are other quantities that depend not merely on their magnitudes, but also on their directions. A vector , denoted v or ~ v , is a quantity that has both magnitude and direction. Velocity Acceleration Force

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Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Representing a Vector A vector is represented as a directed line segment. Its magnitude is represented by the length of the line segment, while its direction is given by the direction of the arrow. Two vectors are equal if and only if they have the same magnitude and direction. A vector may live in any dimension n , though we will focus primarily on n = 2 (a plane) and n = 3, (3-space). P Q
Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Vector Sum Definition The sum of two vectors v and w is the vector v + w formed by the directed line segment from the tail of v to the head of w when the tail of w is placed at the head of v . u v u + v v u Geometrically, the vector v + w is the directed diagonal of the parallelogram defined by the vectors v and w . For this reason, we say we have defined vector addition via the parallelogram law . The zero vector is the vector having magnitude 0. It does not have a direction. We denote the zero vector by 0 . Note 0 + v = v = v + 0 .

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Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Scalar Multiplication Definition We define the scalar multiplication of a vectors v by a scalar λ to be the vector λ v
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