Lecture 12.1: Vectors

University:
The Pennsylvania State University
Course:
Math 231 | Calculus Severable Variables
Academic year:

2024
Views:

464

Pages:

2
Author:

Alexis H.

Definition: a plane is a set of points f(x; y) : x; y 2 Rg Example: the cartesian plane, the complex plane Definition: a vector V is determined by an initial point, P, and a terminal point, Q Example: let P = (0,0) and Q = (1,0) (0; 0) Remark: (0; 1) Definition: two vectors are parallel if the lines through each vector are parallel A vector is not a number. A vector has magnitude and direction. Velocity is a vector; speed is a number Definition: a vector is translated if it is moved without changing magnitude or direction. We say two vectors are equivalent if one is the translation of the other Definition: the components of a vector v = PQ with P = (x1, y1) and Q = (x2, y2) are… a = x2 - x1 (x - component) b = y2 - y1 (y - component) a g ¡¡! ! PQ = v And we denote ! v = hx2 ¡ x1; y2 ¡ y1i Remark: Remark: Two vectors are equivalent if and only if they have the same components Q = (2, 3) and O = (0, 0) are points, not vectors. OQ = <2,3> is a vector Definition: the length of a vector v = is... p jj! v jj = a2 + b2 The length is also called the magnitude Example: find the distance between P = (7,-1) and Q = (4,3) i Create a vector T POT L 3,47 4 7 3 113 2 Find the length of vector v: 11511 FEET V5 5 Vector operations/ vector algebra I Addition: ! ! v = ha; bi w = hc; di ! ! v + w = ha + c; b + di iiw̅ w̅ We add vectors by placing them "tailto-tip" Parallelogram rule: w̅ translated Triangle rule w̅ w̅ E w̅ translated We make a parallelogram where the sides are v and w; the main diagonal is v + w Geometrically: 2 Scalar multiplication: Definition: a scalar is a number 8 2T 4 Example: -3 is a scalar while v = <-3,4> is a vector We can multiply a vector by a scalar using components: a 3 ¡3! v = ¡3 ¢ h¡3; 4i = h9; ¡12i w̅ In general, scalar multiplied by a vector, with v = : Has length of scalar x vector Is in the same direction as v if the scalar is greater than 0 Is in the opposite direction as v if the scalar is less than 0 Is v = <0,0> if the scalar is 0 12 3 Subtraction: T Or T w̅ w̅ i w̅ w̅ w̅ v - w goes from tip to tip Treat like scalar multiplication then addition Definition: a vector is a unit vector if the length of the vector equals 1. We usually replace the narrow with a hat ! v Vector. v b Unit vector Example: what is the unit vector in the direction of v = <2,1> ? In the opposite direction? q p 2 2 jj! v jj = (2) + (1) = 5 So, v is not a unit vector but… ! v ev = ! jj v jj 2 1 ev = h p ; p i 5 5 Definition: the standard basis vectors of all real numbers are i = <1,0> and j = <0,1> Example: <7,3> = 7i + 3j Triangle inequality jj! v +! w jj · jj ! v jj + k! wk Theorem: for any two vectors v and w, With equality if and only if v = 0, w = 0, or if v equals the scalar times w