1. H.Melikyan/12001 Vectors Dr.Hayk Melikyan Departmen of Mathematics and CS melikyan@nccu.edu

2. H.Melikyan/12002 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. 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. P Q Initial point Terminal point Directed Line Segments and Geometric Vectors

3. H.Melikyan/12003 Vector Multiplication If k is a real number and v a vector, the vector k v is called a scalar multiple of the vector v. The magnitude and direction of k v are given as follows: The vector k v has a magnitude of | k | || v ||. We describe this as the absolute value of k times the magnitude of vector v. The vector k v has a direction that is: the same as the direction of v if k > 0, and opposite the direction of v if k < 0

4. H.Melikyan/12004 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. 2. The resultant vector, u + v, extends from the initial point of u to the terminal point of v. Initial point of u u + v v u Resultant vector Terminal point of v The Geometric Method for Adding Two Vectors

5. H.Melikyan/12005 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. v u -u v – u The Geometric Method for the Difference of Two Vectors

6. H.Melikyan/12006 1 1 i j O x y 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. H.Melikyan/12007 Representing Vectors in Rectangular Coordinates Vector v, from (0, 0) to (a, b), is represented as v = a i + b j. 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 a i + b j is called a linear combination of the vectors i and j. The magnitude of v = a i + b j is given by

8. H.Melikyan/12008 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 -5-4-3-212345 5 4 3 2 1 -2 -3 -4 -5 Initial point Terminal point v = -3i + 4j Text Example

9. H.Melikyan/12009 Representing Vectors in Rectangular Coordinates Vector v with initial point P 1 = ( x 1, y 1 ) and terminal point P 2 = ( x 2, y 2 ) is equal to the position vector v = ( x 2 – x 1 ) i + ( y 2 – y 1 ) j. Adding and Subtracting Vectors in Terms of i and j If v = a 1 i + b 1 j and w = a 2 i + b 2 j, then v + w = ( a 1 + a 2 ) i + ( b 1 + b 2 ) j v – w = ( a 1 – a 2 ) i + ( b 1 – b 2 ) j

10. H.Melikyan/120010 If v = 5i + 4j and w = 6i – 9j, find: a. v + w b. v – w. Solution v + w = (5i + 4j) + (6i – 9j) These are the given vectors. = (5 + 6)i + [4 + (-9)]j Add the horizontal components. Add the vertical components. = 11i – 5j Simplify. v + w = (5i + 4j) – (6i – 9j) These are the given vectors. = (5 – 6)i + [4 – (-9)]j Subtract the horizontal components. Subtract the vertical components. = -i + 13j Simplify. Text Example

11. H.Melikyan/120011 Scalar Multiplication with a Vector in Terms of i and j If v = a i + b j and k is a real number, then the scalar multiplication of the vector v and the scalar k is k v = ( ka ) i + ( kb ) j. Example: If v = 2i - 3j, find 5v and -3v

12. H.Melikyan/120012 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 = 0 i + 0 j. Properties of Vector Addition If u, v, and w are vectors, then the following properties are true. Vector Addition Properties 1. u + v = v + u Commutative Property 2. ( u + v ) + w = v + ( u + w ) Associative Property 3. u + 0 = 0 + u = u Additive Identity 4. u + (- u ) = (- u ) + u = 0 Additive Inverse

13. H.Melikyan/120013 Properties of Vector Addition and Scalar Multiplication 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 ( d u ) Associative Property 2. c ( u + v ) = c u + c v Distributive Property 3. ( c + d ) u = c u + d u Distributive Property 4. 1 u = u Multiplicative Identity 5. 0 u = 0 Multiplication Property 6. || c v || = | c | || v ||

14. H.Melikyan/120014 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. H.Melikyan/120015 Definition of a Dot Product If v=a 1 i+b 1 j and w = a 2 i+b 2 j are vectors, the dot product is defined as The dot product of two vectors is the sum of the products of their horizontal and vertical components.

16. H.Melikyan/120016 If v = 5i – 2j and w = -3i + 4j, find: a. v · w b. w · v c. 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 b. w · v = (-3)(5) + (4)(-2) = -15 – 8 = -23 c. v · v = (5)(5) + (-2)(-2) = 25 + 4 = 29 Text Example

17. H.Melikyan/120017 Properties of the Dot Product 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)

18. H.Melikyan/120018 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. H.Melikyan/120019 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. H.Melikyan/120020 Example Find the angle  between v=2i-4j and w=3i+2j. Solution:

21. H.Melikyan/120021 The Dot Product and Orthogonal Vectors Two nonzero vectors v and w are orthogonal if and only if vw=o. Because 0v=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. H.Melikyan/120022 The Vector Projection of v Onto w If v and w are two nonzero vectors, the vector projection of v onto w is If v=3i+4j and w=2i-5j, find the projection of v onto w Solution: Example

23. H.Melikyan/120023 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 v 1 and v 2, where v 1 is parallel to w and v 2 is orthogonal to w. Thus, v = v 1 + v 2. The vectors v 1 and v 2 are called the vector components of v. The process of expressing v as v 1 and v 2 is called the decomposition of v into v 1 and v 2.

24. H.Melikyan/120024 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. H.Melikyan/120025 Definition of Work The work W done by a force F in moving an object from A to B is W = F · AB.