Chapter 12

1. Vectors Chapter 12

2. Objectives • Intro Vectors • Representing Vectors • Algebra and Geometry of Vectors • Cartesian Representation of Vectors in 2-D and 3-D • Properties of Vectors in 2-D and 3-D • Scalar Product of 2 Vectors • Vector Equation of a Line

3. Scalars and Vectors • A scalar is a single number that represents a magnitude • E.g. distance, mass, speed, temperature, etc. • A vectoris a set of numbers that describe both a magnitude and direction • E.g.velocity (the magnitude of velocity is speed), force, momentum, etc. • Can be represented by an arrow v

4. Characteristics of Vectors A Vector is something that has two and only two defining characteristics: • Magnitude: the 'size' or 'quantity' • Direction: the vector is directed from one place to another.

5. z y x Vector Basis • i is the unit vector in the x-direction. • j is the unit vector in the y-direction. • k is the unit vector in the z-direction. 1 1 1 All vectors can be expressed as a linear combination of these 3 vectors

6. B The vector AB A The vector Notation • Using a single small letter with arrow on top. • Using the named points at either end, arrow on top.

7. Notation • Using a “column vector” to show the displacement in the x, y and z (if applicable) directions. • Using “component form” , where i, j and k are unit vectors in the x, y and z directions (equivalent to the column vector form but less easy to use).

8. z y x 3D Vector B 4 A 2 3 This vector can be written as:

9. Calculating Vectors From Points • Point-point subtraction yields a vector • Head point – Tail point v=P-Q

10. point - point = vector B – A B A A – B B A

11. Example • Given the points; A(2, -1, 4), B(5, 0, -3), C(1, 2, 3). Solution:

12. Types of Vectors • Position Vector • Displacement Vector

13. Position Vectors • Position Vectors • Used to define the position of a point. • Always starts at the origin. • The components of the column vector will always be the same as the coordinates of the point.

14. O OR 2D Position Vectors The vector of a point from the origin is called it’s Position Vector R y x Every point has a unique position vector

15. 2D Position Vectors A displacement vector has magnitude and direction, but is not fixed to the origin O OR s A position vector is fixed to the origin R y x

16. 3D Position Vectors z A Similarly in 3D, all points have position vectors a y e.g. The position vector of point a o x

17. Displacement Vectors • Displacement Vectors • Differ from position vectors in that they have no specific position – just represent a change in postion. • A displacement vectoris the difference between an object’s initial and final positions.

18. k k k k Displacement Vectors A Vector has BOTH a Length & a Direction All 4 Vectors here are EQUAL in Length and Travel in SAME Direction. All called k k can be in any position

19. b a d c Describe the vectors in column form

20. ZERO VECTOR • The zero vector, denoted by 0, has length 0. • It is the only vector with no specific direction.

21. Operations with vectors

22. Adding Vectors Add vectors A and B

23. Adding Vectors On a graph, add vectors using the “head-to-tail” rule: Move B so that the head of A touches the tail of B Note: A and B are displacement vectors, “moving” B does not change it. Displacement vectors are only defined by their magnitude and direction, not starting location.

24. Adding Vectors The vector starting at the tail of A and ending at the head of B is C, the sum (or resultant) of A and B.

25. d e e f d c b b c a a f = a + b + c + d +e Addition of multiple Vectors Head-to-tail

26. Vector Subtraction • To subtract vectors b – a, add the negative b + (-a). Note the negative of a vector has the same magnitude and the opposite direction.

27. Vector Subtraction • To subtract vectors b – a, add the negative b + (-a). Note the negative of a vector has the same magnitude and the opposite direction.

28. Vector Subtraction • To subtract vectors b – a, add the negative b + (-a). Note the negative of a vector has the same magnitude and the opposite direction.

29. Vector Subtraction • To subtract vectors b – a, add the negative b + (-a). Note the negative of a vector has the same magnitude and the opposite direction.

30. TRIANGLE LAW You can see why this definition is sometimes called the Triangle Law. All vectors can be added by forming a triangle. The vectors that are head-to-tail are added and the third is the sum.

31. Algebraic Addition and Subtraction • Add or Subtract corresponding components or like terms. Example:

32. SCALAR MULTIPLICATION VECTORS • It is possible to multiply a vector by a scalar • For instance, we want 2v to be the same vector as v + v, which has the same direction as v but is twice as long. • If c is a scalar and v is a vector, the scalar multiplecvis: • The vector whose length is |c| times the length of 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 cv = 0.

33. SCALAR MULTIPLICATION • The definition is illustrated here. • We see that real numbers work like scaling factors here. • That’s why we call them scalars.

34. SCALAR MULTIPLICATION • Notice that two nonzero vectors are parallelif they are scalar multiples of one another.

35. SCALAR MULTIPLICATION • In particular, the vector –v = (–1)v has the same length as v but points in the opposite direction. • We call it the negativeof v.

36. k Write these Vectors in terms of k B D 2k F G ½k 1½k E C -2k A H

37. Summary Vector Operations • Every vector has an inverse • Same magnitude but points in opposite direction • Every vector can be multiplied by a scalar • There is a zero vector • Zero magnitude, undefined orientation • The sum of any two vectors is a vector • Use head-to-tail axiom (triangle law) w v=u+w v v -v u

38. Properties of Vector Arithmetic • If u, v and w are vectors in 2- or 3-space and k and l are scalars, then the following relationships hold. • u + v = v + u • (u + v) + w = u + (v + w) • u + 0 = 0 + u = u • u + (-u) = 0 • k(lu) = (kl)u • k(u + v) = ku + kv • (k + l)u = ku + lu • 1u = u

39. Magnitude of a Vector • The lengthof a vector u is the magnitude or modulus of u and is denoted by |u|. • The magnitude can be found using the Pythagoras’ Theorem in two or three dimensions as appropriate (magnitude vector = square root of the sum of the squares of the components of the vector). • The distance between two points is the magnitude of the vector. • For a constant k, the length of the vector ku = |ku| = |k| |u|.

40. Magnitude of Vector Example Find the length of vector a= 2i+j-3k. Solution

41. Unit Vectors • A unit vector is a vector whose length is 1. • For instance, the basis vectors i, j, and k are all unit vectors. • If a is not the zero vector0, then the unit vector that has the same direction as a and length 1.

42. Example Find the unit vector of 2i – j – 2k. Solution The given vector has length Thus the unit vector with the same direction is ⅓ (2i – j – 2k) = ⅔i - ⅓j - ⅔k or

43. Multiplication of Vectors • By a vector two types Scalar Product (Dot) Vector Product (Cross)

44. Scalar (Dot) Product • The scalar product is a number which can be calculated from two vectors (2-d or 3-d). • The scalar product is used to find the angle between two vectors. • Notation: u·v • Calculating u·v: • also

45. Properties Scalar Product • The dot product obeys many of the properties that hold for ordinary products of real numbers: • An important property is that perpendicular vectors have a dot product equal to zero.

46. Dot Product and Perpendicular Vectors • Two nonzero vectors a and b are perpendicular if the angle between them θ = π/2. • For such vectors we have a ∙ b = |a||b|cos(π/2) = 0 • Conversely, if a ∙ b = 0, then cosθ = 0, soθ = π/2.

47. Dot Product and Perpendicular Vectors • Since the zero vector 0 is considered to be perpendicular to all vectors, we have • Two vectors a and b are perpendicular if and only if a·b = 0. • Further, by properties of the cosine, a ∙ b is • positive for θ < π/2, and • negative for θ > π/2, as the next slide illustrates:

48. Dot Product and Perpendicular Vectors

49. Dot Product and Perpendicular Vectors • We can think of a ∙ b as measuring the extent to which a and b point in the same general direction. • The dot product a ∙ b is… • positive if a and b point in the same general direction, • 0 if they are perpendicular, and • negative if they point in generally opposite directions.

50. Example Show that 2i + 2j – k is perpendicular to5i– 4j + 2k. Solution (2i + 2j – k)∙ (5i– 4j + 2k) = 2(5) + 2(– 4) + (– 1)(2) = 0, these vectors are perpendicular.