chapter 6 vocabulary. section 6.1 vocabulary oblique triangles oblique triangles have no right...

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•a2 = b2 + c2 -2bc Cos (A)•b2 = a2 + c2 -2ac Cos(B)•c2 = a2 + b2 -2ab cos(C)

Heron’s Area FormulaGiven 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 formArea = ½ bh• Oblique TriangleArea = ½ bc sin(A) = ½ ab sin(C) = ½ ac

sin(B)• Heron’s FormulaArea = √[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 isRepresented 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+ v2

2)

• 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

vectoru + 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 vectorku = k <u1, u2> = <ku1, ku2>

Properties of vector addition/scalar multiplicationu and v are vectors. c and d are scalars

1. u + v = v + u 2. ( u + v) + w = u + ( v + w) 3. u + 0 = u4. u + (-u) = 05. c(du) = (cd)u6. (c + d) u = cu + du7. c( u + v) = cu + cv8. 1(u) = u, 0(u) = 09. ||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 · u2. 0 · v = 03. u · (v + w) = u · v + u · w4. 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 componentsForce 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 vectorsAnd 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

nth roots of unity

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