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Geometry of R 2 and R 3

Geometry of R 2 and R 3. Dot and Cross Products. 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. Dot Product in R 3.

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Geometry of R 2 and R 3

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  1. Geometry of R2 and R3 Dot and Cross Products

  2. Dot Product in R2 Let u = (u1, u2) and v = (v1, v2) then the dot product or scalar product, denoted by u.v, is defined as u.v = u1v1 + u2v2

  3. Dot Product in R3 Let u = (u1, u2, u3) and v = (v1, v2, v3) then the dot product or scalar product, denoted by u.v, is defined as u.v = u1v1 + u2v2 + u3v3

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

  5. Theorem 1.2.1 Let u and v be vectors in R2 or R3, and let c be a scalar. Then • u.v = v.u • c(u.v) = (cu).v = u. (cv) • u.(v + w) = u.v + u.w • u.0 = 0 • u.u = ||u||2

  6. Theorem 1.2.2 Let u and v be vectors in R2 or R3, 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. • u = (-1, 2, 3); v = (2, 0, 4) • u = (1, 0, 1); v = (-1, -1, 0)

  8. Orthogonal Vectors Two vectors u and v in R2 or R3 are orthogonal if u.v = 0. Orthogonal, Normal, and Perpendicular, all mean the same.

  9. Theorem 1.2.3 Let u and v be nonzero vectors in R2 or R3 and let  be the angle they form. Then  is • An acute angle if u.v > 0 • A right angle if u.v = 0 • An obtuse angle if u.v < 0

  10. Cross Product (Only in R3 ) Let u = (u1, u2, u3) and v = (v1, v2, v3) be nonzero vectors in R3. Then the cross product, denoted by u x v, is the vector (u2v3 – u3v2, u3v1 – u1v3, u1v2 – u2v1)

  11. Cross Product (Convenient notation) Let u = (u1, u2, u3) and v = (v1, v2, v3) be nonzero vectors in R3. 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 1.2.4 The vector uxv is orthogonal to both u and v.

  14. Theorem 1.2.4 Let u, v, and w be vectors in R3, and let c be a scalar. Then • u x v = –(v x u) • u x (v + w) = (u x v) + (u x w) • (u + v)x w = (u x w) + (v x w) • c(u x v ) = (cu)x v = u x (cv) • u x 0 = 0 x u = 0 • u x u = 0 • ||u x v|| = ||u|| ||v|| sin  = (||u|| ||v|| – ||u.v||2)

  15. Cross Product: Area Let u, and v, be vectors in R3, 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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