Vectors. 1 Basic Definitions. Liming Pang

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1 Vectors Liming Pang 1 Basic Definitions Definition 1. A vector in a line/plane/space is a quantity which has both magnitude and direction. The magnitude is a nonnegative real number and the direction is described by a ray in the line/plane/space. The magnitude of a vector is dented by. Remark. There is a unique vector of magnitude 0, denoted by 0, and we do not assign a specific direction to 0. Example 3. A good example of vectors is the concept of force in physics. When describing a force, we need to know its magnitude (how strong it is) and its direction. Forces of the same magnitude and different directions may have different effects on a particle/object. Example 4. Each real number can be regarded as a vector. The magnitude of a real number is its absolute value, and the direction depends on whether it s positive, negative or zero. Geometric representation of vectors: An oriented line segment from a point A to a point B represents a vector. The length of the line segment represents the magnitude of, and the orientation represents the direction of. A is called the initial point of, and B is called the terminal point of. We also write = AB. B A With the help of geometric representation, we can define the algebraic operations of vectors. 1

2 Definition 5. and are two vectors, positioned in the way that the terminal point of coincides with the initial point of, then define + to be the vector with initial point same as the initial point of and terminal point same as the terminal point of. + Figure 1: Vector Addition Definition 6. and are two vectors, positioned in the way that the initial point of coincides with the initial point of, then define to be the vector with initial point same as the terminal point of and terminal point same as the terminal point of. Figure : Vector Subtraction Definition 7. is a vector. Define to be the vector that has same magnitude with but opposite direction. Remark 8. By the above definitions, we can indeed interpret to be + ( ), as shown by the following picture:

3 + ( ) Figure 3: = + ( ) Definition 9. λ is a real number and is a vector. Define the scalar multiplication λ as follows: λ = λ. The direction of λ is same as that of if λ > 0, the direction of λ is same as that of if λ < 0, and λ = 0 if λ = 0. Figure 4: Scalar Multiplication Remark 10. In particular, we see = ( 1). Definition 11. and are vectors forming an angle θ (0 θ π) when their initial points coincide. Define the dot product of and to be the real number. = cos θ Example 1. If we take the doc product of with itself, θ = 0, we see. = cos 0 = Definition 13. Two nonzero vectors and are perpendicular if the rays representing their directions are perpendicular. Two nonzero vectors and are parallel if the rays representing their directions are parallel. 3

4 Proposition 14. and are two nonzero vectors. They are perpendicular if and only if. = 0, and they are parallel if and only if. = ± Remark 15. When is perpendicular to, we can denote it by Proposition 16. The following rules hold for dot product: 1.. =.. (λ). =.(λ) = λ(.) 3..( + w) =. +. w, ( + w). =. + w. Exercise 17. When =, show that ( + ) ( ) by applying the above propositions. Can you also give a geometric argument? Vectors in Cartesian Coordinates In Cartesian Coordinates (i.e. rectangular coordinates), for any point P = (x, y), we can construct the vector with initial point 0 = (0, 0) and terminal point P, and denote it as OP. We call OP the position vector of the point P. On the other hand, given a vector, we put its initial point coincide with 0 = (0, 0) and its terminal point will be at some point P = (x, y). We then denote = OP = (x, y). Remark 18. We regard two vectors and to be equal if they have the same magnitude and same direction, regardless of their actual positions drawn on the plane. Example 19. In this example, = = OP = (1, ) y P 0 1 x 4

5 The Coordinates bring a convenient description of vector operations: Proposition 0. If = (x 1, y 1 ), = (x, y ), and λ is a real number, then + = (x 1 + x, y 1 + y ), = (x 1 x, y 1 y ), λ = (λx 1, λy 1 ). Proposition 1. If A = (x 1, y 1 ) and B = (x, y ), then the vector AB = (x 1 x, y 1 y ). Proposition. If A = (x 1, y 1 ) and B = (x, y ), then the dot product. = x 1 x + y 1 y Proposition 3. If = (x, y), then = x + y Definition 4. A vector is called a unit vector if = 1. Proposition 5. If is a nonzero vector, then the vector 1 is a unit vector that has the same direction as. We sometimes also write this vector as. Example 6. Let A = (1, 3), B = (, 0). We see AB = (, 0) (1, 3) = (1, 3) Example 7. Let = (1, 4), = (4, 1). Then: + = (1, 4) + (4, 1) = (5, 3), = (1, 4) (4, 1) = ( 3, 5).. = ( 1) = 0, so. Example 8. = (3, 4). = = 5, so 1 = ( 3, 4 ) is a unit vector. The dot product can be applied to compute the angle between vectors: We know by definition. = cos θ, so cos θ =. Example 9. a = ( 3, 1), b = (3, 3). Let s compute the angle between these vectors. a. b = ( 3) = 3. a = ( 3) + 1 =, b = 3 + ( 3) = 3. So cos θ = a. b a b = 3 3 = 1 So cos θ = 1 and 0 θ π, we conclude θ = π 3 5