1. Chapter 6 Vocabulary

2. Section 6.1 Vocabulary

3. Oblique triangles have no right angles.

4. 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

5. Area of an Oblique Triangle
Area = ½ bc sin(A)
= ½ ab sin(C)
= ½ ac sin(B)

6. Section 6.2 Vocabulary

7. Law of Cosines a2 = b2 + c2 -2bc Cos (A) b2 = a2 + c2 -2ac Cos(B)
c2 = a2 + b2 -2ab cos(C)

8. 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

9. 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)]

10. Section 6.3 Vocabulary

11. 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

12. 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.

13. 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

14. 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.

15. 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 >

16. 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>

17. 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||

18. 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

19. 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>

20. 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

21. 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.

22. Section 6.4 Vocabulary

23. 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

24. 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

25. Angle between two vectors
If Ѳ is the angle between two nonzero vectors u and v, then
cos Ѳ = ( u · v) / ||u|| ||v||

26. Definition of orthogonal vectors
The vectors u and v are orthogonal (perpendicular) is u · v = 0

27. Vector components
Force is composed of two orthogonal forces w1 and w2 . F = w1 + w2 w1 and w2 are vector components of F.

28. 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

29. 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

30. Section 6.5 Vocabulary

31. Absolute value of a complex number
The absolute value of the complex number z = a + bi is given by
|a + bi| = √(a2 + b2)

32. 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

33. 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

34. 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)]

35. 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

36. 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

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