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Unformatted text preview: MA 265 LECTURE NOTES: MONDAY, MARCH 24 Inner Product Spaces Length and Direction in R 2 . Recall that V = R 2 is the collection of all 2-vectors: R 2 = v = v 1 v 2 v 1 , v 2 ∈ R . We can define the length of a vector v by drawing a triangle with base v 1 , height v 2 , and hypotenuse we will denote by || v || . The Pythagorean Theorem states v 2 1 + v 2 2 = || v || 2 = ⇒ || v || = q v 2 1 + v 2 2 . On other words, we draw a triangle using the origin (0 , 0) at the tail of the vector, the point ( v 1 , 0) on the x-axis, and the point ( v 1 ,v 2 ) at the head of the vector. Similarly, say that we have two vectors v and u . We can compute the distance between v and u by drawing a triangle using ( u 1 ,u 2 ) at the tail of the vector, the point ( v 1 ,u 2 ), and the point ( v 1 ,v 2 ) at the head of the vector. Upon translating to the origin, we have the following formula: || v- u || = q ( v 1- u 1 ) 2 + ( v 2- u 2 ) 2 . We also define direction. Say that we have a vector x ∈ R 2 . If it makes an angle θ with the x-axis, then we see that x = r cos θ y = r sin θ = ⇒ x = x y = r cos θ sin θ...
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This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.
- Spring '10