# Basic vectors and matrices (12 pages)

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## Basic vectors and matrices

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- Pages:
- 12
- School:
- Massachusetts Institute of Technology
- Course:
- 12 215 - Modern Navigation

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12 215 Modern Navigation Thomas Herring tah mit edu http geoweb mit edu tah 12 215 Review of last class Sextant measurements using the sun We tracked the sun to find its highest elevation and the time this occurs Our one cheat was using computer NTP to get time We will use GPS to check the results 10 14 2009 12 215 Modern Naviation L09 2 1 Today s Class Review of linear Algebra Class will be based on the book Linear Algebra Geodesy and GPS G Strang and K Borre Wellesley Cambridge Press Wellesley MA pp 624 1997 Topics to be covered will be those later in the course General areas are Vectors and matrices Solving linear equations Vector Spaces Eigenvectors and values Rotation matrices 10 14 2009 12 215 Modern Naviation L09 3 Basic vectors and matrices Important basic concepts Vectors A column representing a set of n quantities In two and three dimensions these can be visualized as arrows between points with different coordinates with the vector itself usually having on end at the origin The same concept can be applied to any n dimensional vector Vectors can be added and subtracted head to tail by adding and subtracting the individual components of the vectors Linear combinations of vectors can be formed by scaling and addition The result is another vector e g cv dw Often a bold symbol will be used to denote a vector and some times a line is drawn over the top 10 14 2009 12 215 Modern Naviation L09 4 2 Lengths and dot products The dot product or inner product of two vectors is defined as v w v1w1 v 2 w 2 L v n w n The order of the dot product makes no difference The length or norm of a vector is the square root of the dot product A unit vector is one with unit length If the dot product of two vectors is zero they are said to be orthogonal orthonormal if they are unit vectors The components of a 2 D unit vector are cos and sin of the angle the vector makes to the x axis 10 14 2009 12 215 Modern Naviation L09 5 Angles between vectors The cosine formula v w v w Schwarz

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