Transcript Chapter 6 Vocabulary

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.