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Vector Operations in R2 and R3: Dot and Cross Products

Explore dot and cross products in geometry. Understand scalar products in R2 and R3, with examples and theorems provided. Learn about orthogonal vectors, angles, and cross products in R3. Discover how to calculate areas in parallelograms using vectors.

giovanni-lynagh
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Vector Operations in R2 and R3: Dot and Cross Products

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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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