Chapter 1 Topics in Analytic Geometry
Chapter 1 Topics in Analytic Geometry Chapter 1 Topics in Analytic Geometry
•MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 40F 2F 1 +F 2F 1Physicists and engineers call F 1 +F 2 the resultant of F 1 and F 2 , and they say that theforces F 1 and F 2 are concurrent to indicate that they are applied at the same point.Example 3.12 Suppose that two forces are applied to an eye bracket, as show in Figurebelow. Find the magnitude of the resultant and the angle θ that it makes with the positivex-axis.y‖F 2 ‖ = 300N40 ◦ ‖F 1 ‖ = 200N30 ◦xSolution .........3.3 Dot Product; ProjectionDefinition of the Dot ProductDefinition 3.3 If u = 〈u 1 ,u 2 〉 and v = 〈v 1 ,v 2 〉 are vectors in 2-space, then the dotproduct of u and v is written as u·v and is defined asu·v = u 1 v 1 +u 2 v 2Similarly, if u = 〈u 1 ,u 2 ,u 3 〉 and v = 〈v 1 ,v 2 ,v 3 〉 are vectors in 3-space, then their dotproduct is defined asu·v = u 1 v 1 +u 2 v 2 +u 3 v 3Note that the dot product of two vectors is a scalar. For example,〈4,−3〉·〈3,2〉 = (4)(3)+(−3)(2) = 6〈1,2,−3〉·〈4,−1,2〉 = (1)(4)+(2)(−1)+(−3)(2) = −4
•MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 41Algebraic Properties of the Dot ProductTheorem 3.7 If u, v, and w are vectors in 2-space 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 = ‖v‖ 2(e) 0·v = 0Angle Between VectorsSuppose that u and v are nonzero vectors in 2-space or 3-space that are positioned so theirinitial pointscoincide. Wedefine the angle between u and v tobe theangleθ determinedby the vectors that satisfies the condition 0 ≤ θ ≤ π.uθvuθvuθvuθvTheorem 3.8 If u and v are nonzero vectors in 2-space or 3-space, and if θ is the anglebetween them, thencosθ = u·v(3.5)‖u‖‖v‖Example 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 .........
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•MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 41Algebraic Properties of the Dot ProductTheorem 3.7 If u, v, and w are vectors <strong>in</strong> 2-space 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 = ‖v‖ 2(e) 0·v = 0Angle Between VectorsSuppose that u and v are nonzero vectors <strong>in</strong> 2-space or 3-space that are positioned so their<strong>in</strong>itial po<strong>in</strong>tsco<strong>in</strong>cide. Wedef<strong>in</strong>e the angle between u and v tobe theangleθ determ<strong>in</strong>edby the vectors that satisfies the condition 0 ≤ θ ≤ π.uθvuθvuθvuθvTheorem 3.8 If u and v are nonzero vectors <strong>in</strong> 2-space or 3-space, and if θ is the anglebetween them, thencosθ = u·v(3.5)‖u‖‖v‖Example 3.13 F<strong>in</strong>d 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 F<strong>in</strong>d 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 .........