Chapter 1 Topics in Analytic Geometry

•MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 41Example 3.13 Find the angle between(a) u = 〈4,−3,−1〉 and v = 〈−2,−3,5〉(b) u = −4i+5j+k and v = 2i+3j−7k(c) u = i−2j+2k and v = −3i+6j−6kSolution .........Example 3.14 Find the angle ABC if A = (1,−2,3), B = (2,4,−6), and C = (5,−3,2).A(1,−2,3)θB(2,4,−6)C(5,−3,2)Solution .........Interpreting the Sign of the Dot ProductIt will often be convenient to express Formula (3.5) asu·v = ‖u‖‖v‖cosθ (3.6)which expresses the dot product of u and v in terms of the lengths of these vectors andthe angle between them. Since u and v are assumed to be nonzero vectors, this version ofthe formula make it clear that the sign of u·v is the same as the sign of cosθ. Thus, wecan tell from the dot product whether the angle between two vectors is acute or obtuse orwhether the vectors are perpendicular.uθvuθvuθvu·v > 0u·v < 0u·v = 0Direction AnglesIn both 2-space and 3-space the angle between a nonzero vector v and the vectors i, j,and k are called the direction angles of v, and the cosines of these angles are called thedirection cosines of v. Formulas for the direction cosines of a vector can be obtainedform Formula (3.5). For example, if v = v 1 i+v 2 j+v 3 k, thencosα = v·i‖v‖‖i‖ = v 1 v·j, cosβ =‖v‖ ‖v‖‖j‖ = v 2 v·k, cosγ =‖v‖ ‖v‖‖k‖ = v 3‖v‖