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Vectors

Vectors. Part I. Plan for the day…. Introduction to Vectors Definition Components of a vector Operations with Vectors Unit Vectors Direction Angles Homework. What You Should Learn. Represent vectors as directed line segments. Write the component forms of vectors.

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Vectors

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  1. Vectors Part I

  2. Plan for the day… • Introduction to Vectors • Definition • Components of a vector • Operations with Vectors • Unit Vectors • Direction Angles • Homework

  3. What You Should Learn • Represent vectors as directed line segments. • Write the component forms of vectors. • Perform basic vector operations and represent them graphically. • Write vectors as linear combinations of unit vectors. • Find the direction angles of vectors. • Use vectors to model and solve real-life problems.

  4. Vectors Vectors – directed line segments Initial Point P ( p1, p2 ) Terminal Point Q ( q1, q2 ) Vectors are denoted by lower case bold letters ( u, v, w ) Q (q1, q2) P (p1, p2)

  5. Q (5, 7) S (-3, 5) P (3, 1) R (-5,-1) Vector Components Vectors have two major components Direction Slope Central Angle Magnitude (length) ||v|| =||PQ|| Distance formula Two vectors with the same direction and magnitude are equivalent

  6. More Vector Stuff A vector with its initial point at the origin is said to be in standard position. If a vector, v, is in standard position with its initial point at the origin, then the terminal point is (v1, v2) The component form of a vector, v, is written v = <v1 ,v2> The magnitude of a vector in standard position ||v|| =||PQ|| = Q (q1, q2) (v1, v2) P (p1, p2)

  7. Even More Vector Stuff Two vectors in component form u = <u1 ,u2> and v = <v1 ,v2> Are the same component vector if u1 = v1 and u2 = v2 Zero Vector 0 is the component vector u =<0, 0> Which has both its initial and terminal point at the origin The negative of a vector v is a vector that has the opposite direction of v

  8. Try These Find the component form and magnitude of the vector v. • Initial Point (1, 11), Terminal Point (9,3) • Initial Point (-2, 7), Terminal Point (5, -17)

  9. Vector Operations

  10. Scalar Multiplication If k is positive, kv has the same direction as v, and if k is negative, kv has the direction opposite that of v, as shown Figure 6.19

  11. Vector Addition To add two vectors u and vgeometrically, first position them (without changing their lengths or directions) so that the initial point of the second vector v coincides with the terminal point of the first vector u. The sum u + v is the vector formed by joining the initial point of the first vector u with the terminal point of the second vector v.

  12. Vector Subtaction The negative of v= v1, v2is –v = (–1)v = –v1, –v2 and the difference of u and v is u – v = u + (–v) = u1– v1, u2– v2. u – v = u + (–v)

  13. Vector Operations Scalar Multiplication (see figure 6.19 page 429) Vector Addition (see figure 6.20 page 429) Vector Subtraction (see figure 6.21 page 429) Properties of Operations (see page 431)

  14. Try These u = <2,3> and v = <-1,5> Find:u + vu – 2v 3u2u + 3v

  15. Unit Vectors In many applications of vectors, it is useful to find a unit vector that has the same direction as a given nonzero vector v. A unit vector is a vector that is in the same direction as a given non zero vector but has a magnitude of 1 Therefore… If u is a unit vector in the same direction as v then u is the scalar multiple of v and “u is the unit vector in the direction of v” “u is a vector one unit long and has the same direction as v”

  16. Example – Find a unit vector Find the unit vector in the same direction as v, where v=<-2, 5> Demonstrate that the unit vector has a magnitude of 1

  17. Standard Unit Vectors Horizontal i = <1, 0> Vertical j = <0, 1> If v = <v1 ,v2> then the standard unit vector form of v can be represented as a sum of the two standard unit vectors: v1i + v2j

  18. Unit Vectors These vectors can be used to represent any vectorv = v1, v2, as follows. v = v1, v2 = v11, 0 + v20, 1 = v1i + v2j The scalars v1 and v2 are called the horizontal and vertical components of v, respectively.

  19. Unit Vectors The vector sum v1i + v2j is called a linear combination of the vectors i and j. Any vector in the plane can be written as a linear combination of the standard unit vectors i and j.

  20. Working with Standard Unit Vectors Working with both forms. Write in Standard Unit Vector Form: • <2, 3> • <-1, 4> Write in Component Form • 4i +7j • 2i – 5j

  21. Try These Find the unit vector in the direction of:u = <2,3>v = <2, -1>w = 3i – 2j

  22. Examples • Let u be a vector with an initial point (2, -5) and terminal point (-1, 3)write u in component form and in combination of standard unit vector form. • Let u = -3i + 8j and v = 2i – j, find 2u – 3v.

  23. Direction Angles If u is a unit vector such that  is the angle (measured counterclockwise) from the positive x-axis to u, the terminal point of u lies on the unit circle and you have u = x, y= cos , sin   = (cos  )i + (sin  )j The angle  is the direction angle of the vector u. ||u||= 1

  24. Direction Angles Given Unit Vectoru = <cos , sin> = cos i + sinj Vector “v” = is a scalar multiple of u v = ||v|| <cos , sin> = ||v|| (cos i + sinj) = ||v|| cos i + ||v|| sinj If v = <v1, v2> Example 7 page 433

  25. Example • Find the direction angle of the following vectors:u = 3i + 3j and v = 3i – 4j • Think about the location of the vector - which quadrant is it located?

  26. Try These Find the direction angle:u = <2,3>v = <2, -1>w = -3i – 3j

  27. Homework 37 6.3 Page 436 # 7, 9, 19-25 odd (don’t sketch), 31, 37, 43, 47, 49, 53, 55, 57, 59

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