University:
California State University, Northridge
Course:
MATH 150B | Calculus II
Academic year:

2023
Views:

397

Pages:

5
Author:

Yung D

12.2 Vectors Def. The vector is (25 point directed line segment Als with A initial point Terminal point leng th • Two vectors are equal if they have the same length and direction regardless the initial point" • Vectors are usually written in →lower case and bold face: u, v, w or with arrows: u,,3 = only directed line segment = whose initial point is the origin It is the representative of in stanbul position veetor standard position and is the vector we normally use to represent. Since is a three-dimensional vector with initial point at origin and terminal point (vi,va, Vs), the component form of is = In two dimension is in plane • The magnitude or length of = på is 11 =++= √(xx)²+(3-8) (2-2) Note: The only vector with length o is zero vector 8 = <0, or =<0,0,0 This vector is also the only vecter with no direction. one (26) Ex Find Tas the component form To the length of the vector with Initial point 2(-2,5,2) and terminal point @(-4,3,3) Tas The standard position vechr = på has component form =(4+2,3-5,3-27 =<-2,-2,1 = = √4+4+1 = √9 = 3 <2-2-->=>= Vector Algebra Operations Let = Addition: += <4ith, Us + Ve, Us۲۷۵ Pos Scalar multiplication: Ku = where KEIR is scalar te scale => = The difference The parallelogram law of vector affition (26) Ex Find Tas the component form To the length of the vector with Initial point 2(-2,5,2) and terminal point @(-4,3,3) Tas The standard position vechr = på has component form =(4+2,3-5,3-27 =<-2,-2,1 = = √4+4+1 = √9 = 3 <2-2-->=>= Vector Algebra Operations Let = Addition: += <4ith, Us + Ve, Us۲۷۵ Pos Scalar multiplication: Ku = where KEIR is scalar te scale => = The difference The parallelogram law of vector affition Unit Vectors (28) A vector of length 1 is called a unit vector. The standard unit vectors are 7=<1,0,0),=<0,1,07, <0,0,1) Any vector = <1,, can be written as a linear combination of the standard unit vectors: = = <,,0,0) + 20, V2, 07 +20,0,3> = 4 <1,0,0> + V₂ <0,1, 07+ Vs <0,01 +3 the x-component of the j-component of the i-component of 2 (2) =x²++ PP2=(x2-x)+(-8)+(2-2)尺 +(५-५) R If 8, then 10 and (2) This 50 means V that is called a unit vector in the direction of the direction of 7. En Find a unit vector & in the direction of the recter from 2, (2, 1, 2) to 2, (4,3,1) lul = √4+4+1 = √9 = 3 2j- The unit vector = is the direction of u. Exg Let =37-43 be a velocity vector. Express as a product of its speed times a unit vector in the direction of motion. (29) Speed is the magnitude (length) of given by 131=√9+16= √26 = 5 The unit vector- has the same So 4 direction as v. = 37-4)=5(-3) length "speed" Direction of wation Summary: we can express any vector interms of it's length and direction, by writing = 101 U Est Express the vector 21+-z length = Direction √4+1+4 = √9 =37 = as a product of its length and direction.. 2i+j-2k = 3 (2) + Midpoint of a line Segment: The midpoint t of the line segment jeming the points L.(X, 12.) and le (x, t%, 25) is the point Exe: Find the midpoint of the line segment joining 오, (-1, 1,5), L(2,5,0) Midpoint in the direction of PiP Jahih = (2+) 16-1)+(0-0) = 31+4J-5K The direction is M 4 131=√9+16+25=√50 = 5√2