Elementary Vector Analysis

Elementary Vector Analysis Three-Dimensional Coordinate Systems Right hand rule: to choose an orientation on R3 = R×R×R Curl fingers around the z-axis in the direction off a counterclockwise rotation from the positive x-axis to the positive y-axis. The thumb points up in the direction of the positive z-axis. Coordinate planes The first octant is determined by the positive axes. How to plot a point Move a units along the x-axis then move b units parallel to the y-axis then c units parallel to the z-axis. The Distance Formula Two dimensions: The distance between two points P1 = (x1 , y1 ) and P2 = (x2 , y2 ) is p d = (x2 − x1 )2 + (y2 − y1 )2 Three dimensions: The distance between two points P1 = (x1 , y1 , z1 ) and P2 = (x2 , y2 , z2 ) is p d = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 1 2 Vectors A vector is a quantity that has both a magnitude and a direction. ~ has initial point A and terminal point ~v = AB B. Two vectors with the same length and direction are equivalent, regardless of their position in space, i.e. ~v = ~u. The zero vector ~0 has length 0 and no specific direction. Vector Addition Scalar multiplication If c ∈ R is a scalar and ~v is a vector, the scalar multiple c~v is the vector whose length is |c||~v | and whose direction is the same as ~v if c > 0 and is opposite to ~v if c < 0. If c = 0 or ~v = 0 then c~v = ~0. Two nonzero vectors are parallel if and only if they are scalar multiples of each other. Vector subtraction ~u − ~v = ~u + −~v 3 Vector Components If a vector ~a has its initial point at the origin (0, 0, 0) and its terminal point at P = (a1 , a2 , a3 ), we write ~a = ha1 , a2 , a3 i and call ~a the position vector of the point P. Length or Magnitude of a vector is the distance between its initial and terminal points. Two dimensions: |~a| = Three dimensions: |~a| = p a21 + a22 p a21 + a22 + a23 A unit vector is a vector with length one. The vector ~u = vector ~a. ~a |~a| is a unit vector in the direction of the Vector Algebra: Let ~a = ha1 , a2 , a3 i and ~b = hb1 , b2 , b2 i and c ∈ R. Then (1) ~a + ~b = ha1 + b1 , a2 + b2 , a3 + b3 i (2) ~a − ~b = ha1 − b1 , a2 − b2 , a3 − b3 i (3) c~a = hca1 , ca2 , ca3 i Standard Basis Vectors: ha1 , a2 , a3 i = a1~i + a2~j + a3~k ~i = h1, 0i, ~j = h0, 1i ~i = h1, 0, 0i, ~j = h0, 1, 0i, ~k = h0, 0, 1i Properties of Vectors If ~a, ~b, and ~c are vectors and s and t are scalars, then • • • • ~a + ~b = ~b + ~a ~a + (~b + ~c) = (~a + ~b) + ~c ~a + ~0 = ~a ~a + (−~a) = ~0 • • • • s(~a + ~b) = s~a + s~b (s + t)~a = s~a + t~a (st)~a = s(t~a) 1 · ~a = ~a 4 The Dot Product: ~a = ha1 , a2 , a3 i, ~b = hb1 , b2 , b3 i ~a · ~b = a1 b1 + a2 b2 + a3 b3 Properties of the Dot Product: If ~a, ~b, ~c are vectors and s is a scalar, then (1) (2) (3) (4) (5) ~a · ~a = |~a|2 ~a · ~b = ~b · ~a ~a · (~b + ~c) = ~a · ~b + ~a · ~c (s~a) · ~b = s(~a · ~b) = ~a · (s~b) ~0 · ~a = ~0 Theorem. Let 0 ≤ θ ≤ π be the angle between the vectors ~a and ~b. Then ~a · ~b = |~a||~b| cos θ. Corollary. Two nonzero vectors ~a and ~b are orthogonal (perpendicular) if and only if ~a · ~b = 0. 5 The Cross Product a b = ad − bc. c d The cross product of two vectors ~a = ha1 , a2 , a3 i and ~b = hb1 , b2 , b3 i in R3 is Recall the determinant of a 2 × 2 matrix is ~i ~j ~k ~ ~a × b = a1 a2 a3 b1 b2 b3 = a2 a3 ~ a a a a i − 1 3 ~j + 1 2 ~k b2 b3 b1 b3 b1 b2 = ha2 b3 − a3 b2 , −(a1 b3 − a3 b1 ), a1 b2 − a2 b1 i Geometric Interpretations of the Cross Product (1) ~a × ~b is a vector orthogonal to both ~a and ~b, pointing in the direction determined by the right hand rule. (2) |~a × ~b| = |~a||~b| sin(θ) where θ ∈ [0, π] is the angle between ~a and ~b. (3) |~a × ~b| is the area of the parallelogram with sides ~a and ~b. Corollary. Two nonzero vectors ~a and ~b are parallel if and only if ~a × ~b = ~0. Algebraic Properties of Cross Product: Let ~a, ~b, ~c ∈ V3 and s ∈ R. Then (1) ~a × ~b = −~b × ~a (2) (s~a) × ~b = s(~a × ~b) = ~a × (s~b) (3) ~a × (~b + ~c) = (~a × ~b) + (~a × ~c) (4) (~a + ~b) × ~c = (~a × ~c) + (~b × ~c) (5) ~a · (~b × ~c) = (~a × ~b) · ~c