Chapter 6 Vocabulary Section 6.1 Vocabulary Oblique Triangles •Oblique triangles have no right angles. Law of Sines • If ABC is a triangle with sides a,b, and c then a/ sin(A) = b/sin(B) = c / sin(C) *note: law of sines can also be written in reciprocal form Area of an Oblique Triangle •Area = ½ bc sin(A) = ½ ab sin(C) = ½ ac sin(B) Section 6.2 Vocabulary Law of Cosines 2 •a 2 b 2 c = + -2bc Cos (A) 2 2 2 •b = a + c -2ac Cos(B) 2 2 2 •c = a + b -2ab cos(C) Heron’s Area Formula Given any triangle with sides of lengths a, b, and c, the area of the triangle is given by Area = √[s(s-a)(s-b)(s-c)] Where s = (a + b + c) / 2 Formulas for Area of a triangle • Standard form Area = ½ bh • Oblique Triangle Area = ½ bc sin(A) = ½ ab sin(C) = ½ ac sin(B) • Heron’s Formula Area = √[s(s-a)(s-b)(s-c)] Section 6.3 Vocabulary Directed line segment • To represent quantities that have both a magnitude and a direction you can use a directed line segment like the one below: Terminal Point Initial point Magnitude • Magnitude is the length of a Directed line segment. The magnitude of directed line segment PQ is Represented by ||PQ|| and can be found using the distance formula. Component form of a vector • The component form of a vector with initial point P = (p1, p2) and terminal point Q = (q1, q2) is given by PQ = < q1 - p1 , q2 - p2 > = <v1 , v2> = v Magnitude formula • The length or magnitude of a vector is given by ||v|| = √[ (q1 - p1)2 + (q2 - p2)2] = √( v12+ v22) • If ||v|| = 1, then v is a unit vector • ||v|| = 0 iff v is the zero vector. Vector addition • Let u = <u1, u2> and v = < v1, v2 > be vectors. The sum of vectors u and v is the vector u + v = < u1+ v1, u2 + v2 > Scalar multiplication • Let u = <u1, u2> and v = < v1, v2 > be vectors. And let k be a scalar (a real number). The scalar multiple of k times u is the vector ku = k <u1, u2> = <ku1, ku2> Properties of vector addition/scalar multiplication u and v are vectors. c and d are scalars 1. 2. 3. 4. 5. 6. 7. 8. 9. u+v=v+u ( u + v) + w = u + ( v + w) u+0=u u + (-u) = 0 c(du) = (cd)u (c + d) u = cu + du c( u + v) = cu + cv 1(u) = u, 0(u) = 0 ||cv|| = |c| ||v|| How to make a vector a unit vector If you want to make vector v a unit vector: u = unit vector = v / || v|| = (1/ ||v||) v Note* u is a scalar multiple of v. The vector u has a magnitude of 1 and the same direction as v u is called a unit vector in the direction of v Standard unit vectors • The unit vectors <1,0> and <0,1> are called the standard unit vectors and are denoted by i = <1, 0> and j = <0,1> • Given vector v = < v1 , v2> The scalars v1 and v2 are called the horizontal and vertical components of v, respectively. The vector sum v1i + v2j Is a linear combination of the vectors i and j. Any vector in the plane can be written as a linear combination of unit vectors i and j • Given u is a unit vector such that Ѳ is the angle from the positive x axis to u, and the terminal point lies on the unit circle: U = <x,y> = <cosѲ , sinѲ> = (cosѲ)i + (sinѲ)j The angle Ѳ is the direction angle of the vector u. Section 6.4 Vocabulary Dot product • The dot product of u = <u1, u2> and v = < v1 , v2> is given by u · v = u1 v1 + u2 v2 Note* the dot product yields a scalar Properties of the dot product 1. u · v = v · u 2. 0 · v = 0 3. u · (v + w) = u · v + u · w 2 4. v · v = ||v|| 5. c(u ·v) = cu · v = u · cv Angle between two vectors • If Ѳ is the angle between two nonzero vectors u and v, then • cos Ѳ = ( u · v) / ||u|| ||v|| Definition of orthogonal vectors •The vectors u and v are orthogonal (perpendicular) is u·v=0 Vector components Force is composed of two orthogonal forces w1 and w2 . F = w1 + w2 w1 and w2 are vector components of F. Finding vector components • Let u and v be nonzero vectors And u = w1 + w2 ( note w1 and w2 are orthogonal) w1 = projvu (the projection of u onto v) W 2 = u - w1 Projection of u onto v • Let u and v be nonzero vectors. The projection of u onto v is given by 2 Projvu = [(u · v)/ || v|| ] v Section 6.5 Vocabulary Absolute value of a complex number • The absolute value of the complex number z = a + bi is given by 2 2 |a + bi| = √(a + b ) Trigonometric form of a complex number • The trigonometric form of the complex number z = a + bi is given by Z = r (cosѲ + i sinѲ) Where a = rcos Ѳ, and b = rsin Ѳ, r = √(a2 + b2) , and tan Ѳ = b/a The number r is the modulus of z, and Ѳ is called an argument of z Product and quotient of two complex numbers Let z1 = r1(cosѲ1 + i sin Ѳ1 ) and z2 = r2(cosѲ2 + i sin Ѳ2 ) be complex numbers. z1 z2 = r1r2[cos(Ѳ1 + Ѳ2) + i sin (Ѳ1 + Ѳ2) ] z1 /z2 = r1/r2 [cos(Ѳ1 - Ѳ2) + i sin (Ѳ1 - Ѳ2) ], z2 ≠ 0 DeMoivre’s Theorem • If z = r (cosѲ + i sinѲ) is a complex number and n is a positive integer, then zn = [r (cosѲ + i sinѲ)]n n = [r (cos nѲ + i sin nѲ)] Definition of an nth root of a complex number • The complex number u = a + bi is an nth root of the complex number z if Z= n u = (a + bi) n Nth roots of a complex number • For a positive integer n, the complex number\ z = r( cos Ѳ + i sin Ѳ) has exactly n distinct nth roots given by r1/n ( cos([Ѳ + 2∏k]/n) + i sin ([Ѳ + 2∏k]/n) Where k = 0,1,2,…, n-1 nth roots of unity •The n distinct roots of 1 are called the nth roots of unity.