Chapter 6 Vocabulary

Section 6.1 Vocabulary

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 a2 = b2 + c2 -2bc Cos (A) b2 = a2 + c2 -2ac Cos(B) c2 = a2 + b2 -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 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 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 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) W2 = u - w1

Projection of u onto v Let u and v be nonzero vectors. The projection of u onto v is given by Projvu = [(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| = √(a2 + b2)

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 = [rn (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 = un = (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

The n distinct roots of 1 are called the nth roots of unity.