Lecture notes
Lecture notes ECON10071A
- Institution
- Lancaster University (LU)
This document gives an in depth summary of the topic of vectors in multivariable calculus
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Multivariable Calculus15 November 2023 19:42
Real Numbers
• Real numbers can be positive, zero, or negative, and they can be rational or irrational.
• Denote the set of all real numbers as R.
Representation in 1 Dimension (R)
• Real numbers can be thought of as points on a line when an origin and unit length are
chosen.
• Real numbers are also referred to as scalars.
Representation in 2 Dimensions (R2)
• Pairs of real numbers ((x1,x2)) form elements of R2.
• These elements label points in a plane when an origin and orthogonal axes are chosen.
• These elements are also called 2-dimensional vectors and can be represented as arrows.
Representation in 3 Dimensions (R3)
• Triples of real numbers ((x1,x2,x3)) form elements of R3.
• These elements label points in space when an origin and three orthogonal axes are chosen.
• These elements are 3-dimensional vectors.
Generalizing to Rn
• Rn represents all n-tuples of real numbers ((x1,x2,...,xn).
• These elements are points in n-dimensional space and are called n-dimensional vectors.
Vector Operations
• A scalar (α) can multiply a vector (x) to produce αx.
• Vectors (x and y) can be added (x + y) or subtracted (x − y).
• These operations define a vector space.
Geometric Interpretation
• Scalar multiplication (αx) changes the length and direction of a vector.
• Vector addition (x + y) can be visualized as completing a parallelogram or connecting
the heads of vectors.
• Vector subtraction (x − y) can be visualized as completing a parallelogram or
connecting the heads of vectors.
Advanced Math Page 1
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