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Dot Product and Projections in 2D/3D Space

Learn about the dot product, angle between vectors, orthogonal vectors, properties of the dot product, and orthogonal projections in 2D/3D space.

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Dot Product and Projections in 2D/3D Space

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  1. Section 3.3 Dot Product; Projections

  2. THE DOT PRODUCT If u and v are vectors in 2- or 3-space and θis the angle between u and v, then the dot product or Euclidean inner product u∙v is defined by

  3. COMPONENT FORM OF THE DOT PRODUCT If u = (u1, u2) and v = (v1, v2) are vectors in 2-space, then u∙ v = u1v1 + u2v2 If u = (u1, u2, u3) and v = (v1, v2, v3) are vectors in 3-space, then u∙ v = u1v1 + u2v2 + u3v3

  4. ANGLE BETWEEN VECTORS If u and v are nonzero vectors and θis the angle between them, then the angle between them is

  5. DOT PRODUCT, NORM, AND ANGLES Theorem 3.3.1: Let u and v be vectors in 2- or 3-space. (a) v∙ v = ||v||2; that is, ||v|| = (v ∙ v)1/2 (b) If the vectors u and v are nonzero and θ is the angle between them, then θ is acute if and only if u∙ v > 0 θ is obtuse if and only if u ∙ v < 0 θ = π/2 if and only if u ∙ v = 0

  6. ORTHOGONAL VECTORS Perpendicular vectors are also called orthogonal vectors. Two nonzero vectors are orthogonal if and only if their dot product is zero. We consider the zero vector orthogonal to all vectors. To indicate that u and v are orthogonal, we write

  7. PROPERTIES OF THE DOT PRODUCT If u, v, and w are vectors is 2- or 3-space and k is a scalar, then: (a) u∙v = v ∙ u (b) u ∙ (v + w) = u ∙ v + u ∙ w (c) k(u ∙ v) = (ku) ∙ v = u ∙ (kv) (d) v ∙ v > 0 if v≠0, and v ∙ v = 0 if v = 0

  8. ORTHOGONAL PROJECTION Let u and a≠0 be positioned so that their initial points coincide at a point Q. Drop a perpendicular from the tip of u to the line through a, and construct the vector w1 from Q to the foot of this perpendicular. Next form the difference w2 = u− w1. The vector w1 is called the orthogonal projection of u on a or sometimes the vector component of u along a. It is denoted by proja u. The vector w2 is called the vector component of u orthogonal toa and can be written as w2 = u− proja u.

  9. THEOREM Theorem 3.3.3: If u and a are vectors in 2-space or 3-space and if a≠ 0, then (vector component of u along a) (vector component of u orthogonal to a)

  10. NORM OF THE ORTHOGONAL PROJECTION OF u ALONG a

  11. DISTANCE BETWEEN A POINT AND A LINE The distance D between a point P(x0, y0) and the line ax + by + c = 0 is given by the formula

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