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chapter1 - Math 152 Linear Systems Winter 2004 1 Vectors...

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Math 152: Linear Systems – Winter 2004 1: Vectors and Geometry Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Multiplication by a number and vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Co-ordinates .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Properties of vector addition and scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Length of a vector .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Projections .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The determinant in two and three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 The cross product and rotational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 The triple product and the determinant in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Describing linear sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Lines in two dimensions: Parametric form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Lines in two dimensions: Equation form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Lines in three dimensions: Parametric form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Lines in three dimensions: Equation form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Planes in three dimensions: Parametric form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Planes in three dimensions: Equation form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Application: 3-D graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Description of points and the geometry of solutions to systems of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Describing the whole plane in two dimensions and all of space in three dimensions . . . . . . . . . . . . . . . . . . . . . . 24 Linear dependence and independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 c 2003 Richard Froese. Permission is granted to make and distribute copies of this document provided the copyright notice and this permission notice are preserved on all copies.
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2 Math 152: Vectors and Geometry Vectors Vectors are used to describe quantities that have both a magnitude and a direction. You are probably familiar with vector quantities in two and three dimensions, such as forces and velocities. Later in this course we will see that vectors can also describe the configuration of a mechanical system of weights and springs, or the collections of voltages and currents in an electrical circuit. These more abstract vector quantities are not so easily visualized since they take values in higher dimensional spaces. We begin this course by discussing the geometry of vectors in two and three dimensions. In two and three dimensions, vectors can be visualized as arrows. Before we can draw a vector, we have to decide where to place the tail of the vector. If we are drawing forces, we usually put the tail of the vector at the place where the force is applied. For example, in the diagram the forces acting on a pendulum bob are gravity and the restraining force along the shaft. Forces acting on a pendulum If we are drawing the velocity of a particle at a given time, we would place the tail of the velocity vector v ( t ) at the position of the particle at that time. v r(t) (t) Position and velocity of a particle Once we have chosen a starting point for the tails of our vectors (i.e., an origin for space), every point in space corresponds to exactly one vector, namely the vector whose tail is at the origin and whose head is at the given point. For example, in the diagram above, we have chosen an arbitrary point as the origin (marked with a circle) and identified position of the particle with the vector r ( t ).
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Math 152: Vectors and Geometry 3 Multiplication by a number and vector addition There are two basic operations defined for vectors. One is multiplication of a vector by a number (also called scalar multiplication). The other is addition of two vectors.
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