Presentation on theme: "Chapter 6 Vocabulary. Section 6.1 Vocabulary Oblique Triangles Oblique triangles have no right angles."— Presentation transcript:
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 a 2 = b 2 + c 2 -2bc Cos (A) b 2 = a 2 + c 2 -2ac Cos(B) c 2 = a 2 + b 2 -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: Initial point Terminal 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 = (p 1, p 2 ) and terminal point Q = (q 1, q 2 ) is given by PQ = = = v
Magnitude formula The length or magnitude of a vector is given by ||v|| = √[ (q 1 - p 1 ) 2 + (q 2 - p 2 ) 2 ] = √( v 1 2 + v 2 2 ) If ||v|| = 1, then v is a unit vector ||v|| = 0 iff v is the zero vector.
Vector addition Let u = and v = be vectors. The sum of vectors u and v is the vector u + v =
Scalar multiplication Let u = and v = be vectors. And let k be a scalar (a real number). The scalar multiple of k times u is the vector ku = k =
Properties of vector addition/scalar multiplication u and v are vectors. c and d are scalars 1.u + v = v + u 2.( u + v) + w = u + ( v + w) 3.u + 0 = u 4.u + (-u) = 0 5.c(du) = (cd)u 6.(c + d) u = cu + du 7.c( u + v) = cu + cv 8.1(u) = u, 0(u) = 0 9.||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 and are called the standard unit vectors and are denoted by i = and j =
Given vector v = The scalars v 1 and v 2 are called the horizontal and vertical components of v, respectively. The vector sum v 1 i + v 2 j 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 = = = (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 = and v = is given by u · v = u 1 v 1 + u 2 v 2 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 4. v · v = ||v|| 2 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 w 1 and w 2. F = w 1 + w 2 w 1 and w 2 are vector components of F.
Finding vector components Let u and v be nonzero vectors And u = w 1 + w 2 ( note w 1 and w 2 are orthogonal) w 1 = proj v u (the projection of u onto v) W 2 = u - w 1
Projection of u onto v Let u and v be nonzero vectors. The projection of u onto v is given by Proj v u = [(u · v)/ || v|| 2 ] v
Section 6.5 Vocabulary
Absolute value of a complex number The absolute value of the complex number z = a + bi is given by |a + bi| = √(a 2 + b 2 )
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 = √(a 2 + b 2 ), 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 z 1 = r 1 (cosѲ 1 + i sin Ѳ 1 ) and z 2 = r 2 (cosѲ 2 + i sin Ѳ 2 ) be complex numbers. z 1 z 2 = r 1 r 2 [cos(Ѳ 1 + Ѳ 2 ) + i sin (Ѳ 1 + Ѳ 2 ) ] z 1 /z 2 = r 1 /r 2 [cos(Ѳ 1 - Ѳ 2 ) + i sin (Ѳ 1 - Ѳ 2 ) ], z 2 ≠ 0
DeMoivre’s Theorem If z = r (cosѲ + i sinѲ) is a complex number and n is a positive integer, then z n = [r (cosѲ + i sinѲ)] n = [r n (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 = u n = (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 r 1/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.