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Geometry of R 2 and R 3 Dot and Cross Products. R 2 Dot Product in R 2 Let u = (u 1, u 2 ) and v = (v 1, v 2 ) then the dot product or scalar product,

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Presentation on theme: "Geometry of R 2 and R 3 Dot and Cross Products. R 2 Dot Product in R 2 Let u = (u 1, u 2 ) and v = (v 1, v 2 ) then the dot product or scalar product,"— Presentation transcript:

1 Geometry of R 2 and R 3 Dot and Cross Products

2 R 2 Dot Product in R 2 Let u = (u 1, u 2 ) and v = (v 1, v 2 ) then the dot product or scalar product, denoted by u. v, is defined as u. v = u 1 v 1 + u 2 v 2

3 R 3 Dot Product in R 3 Let u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ) then the dot product or scalar product, denoted by u. v, is defined as u. v = u 1 v 1 + u 2 v 2 + u 3 v 3

4 Example Find the dot product of each pair of vectors a. u = (-3, 2, -1); v = (-4, -3, 0) b. u = (-4, 0, -2); v = (-3, -7, 6) c. u = (-6, 3); v = (5, -8)

5 Theorem R 2 R 3 Let u and v be vectors in R 2 or R 3, and let c be a scalar. Then a. u. v = v. u b. c(u. v) = (cu). v = u. (cv) c. u. (v + w) = u. v + u. w d. u. 0 = 0 e. u. u = ||u|| 2

6 Theorem R 2 R 3 Let u and v be vectors in R 2 or R 3, and let  be the angle they form. Then u. v = ||v|| ||u|| cos  If u and v are nonzero vectors, then

7 Example Find the angle between each pair of vectors. a. u = (-1, 2, 3); v = (2, 0, 4) b. u = (1, 0, 1); v = (-1, -1, 0)

8 Orthogonal Vectors R 2 R 3 Two vectors u and v in R 2 or R 3 are orthogonal if u. v = 0. Orthogonal, Normal, and Perpendicular, all mean the same.

9 Theorem R 2 R 3 Let u and v be nonzero vectors in R 2 or R 3 and let  be the angle they form. Then  is a.An acute angle if u. v > 0 b.A right angle if u. v = 0 c.An obtuse angle if u. v < 0

10 R 3 Cross Product (Only in R 3 ) R 3 Let u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ) be nonzero vectors in R 3. Then the cross product, denoted by u x v, is the vector (u 2 v 3 – u 3 v 2, u 3 v 1 – u 1 v 3, u 1 v 2 – u 2 v 1 )

11 Cross Product (Convenient notation ) R 3 Let u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ) be nonzero vectors in R 3. Then u x v, is the vector obtained by evaluating the determinant:

12 Example Find the cross product of the following vectors u = (-1, 1, 0); v = (2, 3, -1)

13 Theorem The vector uxv is orthogonal to both u and v.

14 Theorem R 3 Let u, v, and w be vectors in R 3, and let c be a scalar. Then a. u x v = –(v x u) b. u x (v + w) = (u x v) + (u x w) c. (u + v) x w = (u x w) + (v x w) d. c(u x v ) = (cu) x v = u x (cv) e. u x 0 = 0 x u = 0 f. u x u = 0 g. ||u x v|| = ||u|| ||v|| sin  =  (||u|| ||v|| – ||u. v|| 2 )

15 Cross Product: Area R 3 Let u, and v, be vectors in R 3, Then the area of the parallelogram determined by u and v is ||u x v|| = ||u|| ||v|| sin 

16 Example Find the area of the parallelogram determined by the vectors u = (-1, 1, 0) and v = (2, 3, -1).

17 Homework 1.2


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