Pc8-2 Vectors2 Notes
- 1. Precalculus 8-2 Algebraic Vectors Vectors can be used algebraically using _______________________________________. • For example, 〈3, 4〉 can represent a vector in standard position, with the __________________ point at the origin and the terminal point at ____________. Magnitude can be found using the ___________________________________________. • Drawing the horizontal and vertical ____________________ of the vector yields a ________________________________. • For example, the magnitude of 〈3, 4〉 is ______________________. Representing a Vector as an Ordered Pair Initial: P1(x1, y1) Terminal: P2(x2, y2) • The ordered pair representing P1P2 is ________________________ • Its magnitude |P1P2| = (x2 − ξ1 )2 + ( ψ − ψ )2 2 1 Example: Write the ordered pair that represents the vector from X(-3, 5) to Y(4, -2). Then find the magnitude of XY. Vector Operations • Addition: _________________________________________________________ r r r Example: m = 〈5, -7〉 n = 〈0, 4〉, and p = 〈-1, 3〉 r r Find m + p • Subtraction: _______________________________________________________ r r Example: Find m – n • Scalar Multiplication: ________________________________________________ r r r r Examples: Find 7 p Find 2 m + 3 n – p
- 2. A vector that has a magnitude of one is called a _________________________________. • The unit vector in the direction of the _______________________________ is always represented _______. It has the coordinates _____________. • The unit vector in the direction of the positive _____________ is always represented ______. It has the coordinates ______________. r Any vector a = α1 , α2 can be expressed as a _______________ of unit vectors. r Proof: a = α1 + α2 ι ϕ Note: The _____________ vector, which has coordinates ___________, is denoted ____. Examples: Write AB as the sum of unit vectors for A(4, -1) and B(6, 2). Write DE as the sum of unit vectors for D(2, -7) and E(-1, 5). HW p. 497 (14 – 40 even)