Vectors

1. Vectors Dr .Hayk Melikyan Departmen of Mathematics and CS melikyan@nccu.edu

2. A line segment to which a direction has been assigned is called a directed line segment. The figure below shows a directed line segment form P to Q. We call P the initial point and Q the terminal point. We denote this directed line segment by PQ. Q Initial point Terminal point P The magnitude of the directed line segment PQ is its length. We denote this by || PQ ||. Thus, || PQ || is the distance from point P to point Q. Because distance is nonnegative, vectors do not have negative magnitudes. Geometrically, a vector is a directed line segment. Vectors are often denoted by a boldface letter, such as v. If a vector v has the same magnitude and the same direction as the directed line segment PQ, we write v = PQ. Directed Line Segments and Geometric Vectors

3. If k is a real number and v a vector, the vector kv is called a scalar multiple of the vector v. The magnitude and direction of kv are given as follows: The vector kv has a magnitude of |k| ||v||. We describe this as the absolute value of k times the magnitude of vector v. The vector kv has a direction that is: the same as the direction of v if k > 0, and opposite the direction of v if k < 0 Vector Multiplication

4. Resultant vector u + v Terminal point of v v u Initial point of u A geometric method for adding two vectors is shown below. The sum of u + v is called the resultant vector. Here is how we find this vector. 1. Position u and v so the terminal point of u extends from the initial point of v. The Geometric Method for Adding Two Vectors 2. The resultant vector, u + v, extends from the initial point of u to the terminal point of v.

5. -u v – u v u -u The difference of two vectors, v – u, is defined as v – u = v + (-u), where –u is the scalar multiplication of u and –1: -1u. The difference v – u is shown below geometrically. The Geometric Method for the Difference of Two Vectors

6. y 1 j x i O 1 The i and j Unit Vectors Vector i is the unit vector whose direction is along the positive x-axis. Vector j is the unit vector whose direction is along the positive y-axis.

7. Vector v, from (0, 0) to (a, b), is represented as v = ai + bj. The real numbers a and b are called the scalar components of v. Note that a is the horizontal component of v, and b is the vertical component of v. The vector sum ai + bj is called a linear combination of the vectors i and j. The magnitude of v = ai + bj is given by Representing Vectors in Rectangular Coordinates

8. Terminal point 5 4 3 v = -3i + 4j 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 Initial point -4 -5 Sketch the vector v = -3i + 4j and find its magnitude. Solution For the given vector v = -3i + 4j, a = -3 and b = 4. The vector, shown below, has the origin, (0, 0), for its initial point and (a, b) = (-3, 4) for its terminal point. We sketch the vector by drawing an arrow from (0, 0) to (-3, 4). We determine the magnitude of the vector by using the distance formula. Thus, the magnitude is Text Example

9. Representing Vectors in Rectangular Coordinates • Vector v with initial point P1 = (x1, y1) and terminal point P2 = (x2, y2) is equal to the position vector v = (x2 – x1)i + (y2 – y1)j. Adding and Subtracting Vectors in Terms of i and j If v = a1i + b1j and w = a2i + b2j, then v + w = (a1 + a2)i + (b1 + b2)j v – w = (a1 – a2)i + (b1 – b2)j

10. Solution • v + w = (5i + 4j) + (6i – 9j) These are the given vectors. • = (5 + 6)i + [4 + (-9)]jAdd the horizontal components. Add thevertical components. • = 11i – 5jSimplify. If v = 5i + 4j and w = 6i – 9j, find: a. v + wb. v – w. Text Example • v + w = (5i + 4j) – (6i – 9j) These are the given vectors. • = (5 – 6)i + [4 – (-9)]jSubtract the horizontal components. • Subtractthe vertical components. • = -i + 13jSimplify.

11. Scalar Multiplication with a Vector in Terms of i and j • If v = ai + bj and k is a real number, then the scalar multiplication of the vector v and the scalar k is • kv = (ka)i + (kb)j. Example: If v = 2i - 3j, find 5v and -3v

12. The Zero Vector The vector whose magnitude is 0 is called the zero vector, 0. The zero vector is assigned no direction. It can be expressed in terms of I and j using • 0 = 0i + 0j. Properties of Vector Addition If u, v, and w are vectors, then the following properties are true. Vector Addition Properties 1. u + v = v + uCommutative Property 2. (u + v) + w = v + (u + w) Associative Property 3. u + 0 = 0 + u = uAdditive Identity 4. u + (-u) = (-u) + u = 0Additive Inverse

13. If u, v, and w are vectors, and c and d are scalars, then the following properties are true. Scalar Multiplication Properties 1. (cd)u = c(du) Associative Property 2. c(u + v) = cu + cvDistributive Property 3. (c + d)u = cu + duDistributive Property 4. 1u = uMultiplicative Identity 5. 0u = 0Multiplication Property 6. ||cv|| = |c| ||v|| Properties of Vector Addition and Scalar Multiplication

14. Finding the Unit Vector that Has the Same Direction as a Given Nonzero Vector v For any nonzero vector v, the vector is a unit vector that has the same direction as v. To find this vector, divide v by its magnitude. Example Find a unit vector in the same direction as v=4i-7j

15. Definition of a Dot Product The dot product of two vectors is the sum of the products of their horizontal and vertical components. If v=a1i+b1j and w = a2i+b2j are vectors, the dot product is defined as

16. If v = 5i – 2j and w = -3i + 4j, find: a. v · wb. w · vc. v · v. Solution To find each dot product, multiply the two horizontal components, and then multiply the two vertical components. Finally, add the two products. a. v · w = 5(-3) + (-2)(4) = -15 – 8 = -23 Text Example b. w · v = (-3)(5) + (4)(-2) = -15 – 8 = -23 c. v · v = (5)(5) + (-2)(-2) = 25 + 4 = 29

17. If u, v, and w, are vectors, and c is a scalar, then 1. u · v = v · u 2. u · (v + w) = u · v + u · w 3. 0 · v = 0 4. v · v = || v ||2 5. (cu) · v = c(u · v) = u · (cv) Properties of the Dot Product

18. Alternative Formula for the Dot Product • If v and w are two nonzero vectors and  is the smallest nonnegative angle between them, then v · w = ||v|| ||w|| cos.

19. Formula for the Angle between Two Vectors If v and w are two nonzero vectors and  is the smallest nonnegative angle between v and w, then

20. Example Solution: Find the angle  between v=2i-4j and w=3i+2j.

21. The Dot Product and Orthogonal Vectors Two nonzero vectors v and w are orthogonal if and only if v•w=o. Because 0•v=0, the zero vector is orthogonal to every vector v. Example Are the vectors v=3i-2j and w=3i+2j orthogonal? The vectors are not orthogonal.

22. The Vector Projection of v Onto w If v and w are two nonzero vectors, the vector projection of v onto w is Example If v=3i+4j and w=2i-5j, find the projection of v onto w Solution:

23. The Vector Components of v Let v and w be two nonzero vectors. Vector v can be expressed as the sum of two orthogonal vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. Thus, v = v1 + v2. The vectors v1 and v2 are called the vector components of v. The process of expressing v as v1 and v2 is called the decomposition of v into v1 and v2.

24. Example Let v=3i+j and w=2i-3j. Decompose v into two vectors, where one is parallel to w and the other is orthogonal to w. Solution:

25. Definition of Work • The work W done by a force F in moving an object from A to B is • W = F · AB.