Section 3.3 Dot Product; Projections

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

COMPONENT FORM OF THE DOT PRODUCT If u = (u 1, u 2 ) and v = (v 1, v 2 ) are vectors in 2- space, then u ∙ v = u 1 v 1 + u 2 v 2 If u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ) are vectors in 3-space, then u ∙ v = u 1 v 1 + u 2 v 2 + u 3 v 3

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

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 acuteif and only ifu ∙ v > 0 θ is obtuseif and only ifu ∙ v < 0 θ = π/2if and only ifu ∙ v = 0

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

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

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 w 1 from Q to the foot of this perpendicular. Next form the difference w 2 = u − w 1. The vector w 1 is called the orthogonal projection of u on a or sometimes the vector component of u along a. It is denoted by proj a u. The vector w 2 is called the vector component of u orthogonal to a and can be written as w 2 = u − proj a u.

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)

NORM OF THE ORTHOGONAL PROJECTION OF u ALONG a

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