2331_Notes_1o2_fill - Math 2331 Linear Algebra Section 1.2...

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Math 2331 Linear Algebra Section 1.2 Dot Product, Length and Angle Between 2 Vectors Given column vectors v and w (with the same number of rows, 11 2 , ... ... nn vw ⎛⎞ ⎛ ⎞ ⎜⎟ ⎜ ⎟ == ⎝⎠ ⎝ ⎠ 2 ) Then 2 2 ... vw vw vw •= + ++ Rules for the dot product 1. for all vectors v and w wv •=• 2. () uvw uvuw •+ =•+• 3. for any scalar k ( uk w kuw Also notice Why? 0 uu •≥ The Length of a Vector v - 1/2 vv v v =• = • v Example - Find the lengths of the following vectors - 1 0 3 4 v =
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1 2 1 2 1 2 v ⎛⎞ ⎜⎟ = ⎝⎠ Component wise - 22 12 ... n vv v v =+ + + 2 This corresponds to our normal understanding of the length of a vector in 2- or 3- dimensional space x y z
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If the length of a vector is 1, it is called a UNIT VECTOR Standard unit vectors in 2-D and 3-D space Nonstandard unit vector Unit vector in the direction of v - v v Example 1 1 1 v ⎛⎞ ⎜⎟ = ⎝⎠
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The Angle Between 2 Vectors - Perpendicular Vectors 2 Vectors are perpendicular if their dot product is equal to 0 The "fancy" term for perpendicular is orthogonal (and this carries the idea of
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This note was uploaded on 08/10/2011 for the course MATH 2331 taught by Professor Staff during the Spring '08 term at University of Houston.

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2331_Notes_1o2_fill - Math 2331 Linear Algebra Section 1.2...

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