Post on 01-Apr-2015
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.