11 Directed line segmentTo represent quantities that have both a magnitude and a direction you can use a directed line segment like the one below:Terminal PointInitial point
12 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.
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 byPQ = < 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 additionLet u = <u1, u2> and v = < v1, v2 > be vectors.The sum of vectors u and v is the vectoru + 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 vectorku = 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 = uu + (-u) = 0c(du) = (cd)u(c + d) u = cu + duc( u + v) = cu + cv1(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 vectorsThe unit vectors <1,0> and <0,1> are called the standard unit vectors and are denoted byi = <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 sumv1i + v2jIs 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)jThe angle Ѳ is the direction angle of the vector u.
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 byZ = r (cosѲ + i sinѲ)Where a = rcos Ѳ, and b = rsin Ѳ, r = √(a2 + b2) , and tan Ѳ = b/aThe 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 TheoremIf z = r (cosѲ + i sinѲ) is a complex number and n is a positive integer, thenzn = [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 ifZ = 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 byr1/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.