Vectors. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) Vectors Spring /

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1 Vectors Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Vectors Spring / 18

2 Introduction - Definition Many quantities we use in the sciences such as mass, volume, distance, can be expressed by a number Such quantities are called scalar-valued There are other quantities for which more than a number is needed For example, to describe the motion of an object, we need to know the velocity of the object as well as the direction in which the object is moving Such quantities are called vector-valued Therefore, loosely speaking, a vector is a quantity which has both magnitude and direction Another way to think of a vector is that it represents a displacement We now give a more precise definition Definition A 2-D vector is an ordered pair a, b A 3-D vector is an ordered triple a, b, c In general, an n-vector is an ordered n-tuple a 1, a 2,, a n These numbers are called the coordinates or the components of the vector Philippe B Laval (KSU) Vectors Spring / 18

3 Vectors Versus Points Though vectors seem to have coordinates like points, they do not represent the same thing A point specifies a location In contrast, a vector specifies a displacement A 2-D vector whose coordinates are a, b indicates a displacement of a units in the x direction and b units in the y direction Thus, it is easy to see that if the starting point P has coordinates (x, y) then, the terminal point Q of this vector of coordinates a, b will have coordinates (x + a, y + b) The vector is then represented by an arrow between the two points Conversely, given two points P (x 1, y 1 ) and Q (x 2, y 2 ), then PQ = x 2 x 1, y 2 y 1 Philippe B Laval (KSU) Vectors Spring / 18

4 Vectors Versus Points Similarly, a 3-D vector whose coordinates are a 1, a 2, a 3 indicates a displacement of a 1 units in the x direction, a 2 units in the y direction and a 3 units in the z direction Thus, it is easy to see if the starting point P of this vector has coordinates (x, y, z) then the end point Q of this vector will be (x + a 1, y + a 2, z + a 3 ) The vector is then represented by an arrow between the two points Conversely, given two points P (x 1, y 1, z 1 ) and Q (x 2, y 2, z 2 ), the vector starting at P and ending at Q, denoted PQ, has coordinates PQ = x 2 x 1, y 2 y 1, z 2 z 1 Finally, let us note that if a point P has coordinates (x, y) then the vector OP has coordinates x, y If P has coordinates (x, y, z) then the vector OP has coordinates x, y, z Philippe B Laval (KSU) Vectors Spring / 18

5 Vectors Versus Points Example If we displace a point P (2, 4) along a vector 1, 5, what are the coordinates of the terminal point of the vector? Example Find the coordinates of the vector PQ where P = (2, 4) and Q = (3, 9) Example Find the coordinates of the vector AB where A = (1, 2, 3) and B = (3, 1, 7) Philippe B Laval (KSU) Vectors Spring / 18

6 Equivalent Vectors A vector will have many different representation, depending on which starting point we selected All these representations represent the same vector Because position is not part of the definition of a vector, only direction and magnitude count This means that as long as two vectors have the same magnitude and direction, we consider them the same, even if they are not in the same position We say they are equivalent Unless we need to draw a vector in a specific position, we will draw our vectors starting at the origin Finally, let us note that two vectors are equal if and only if their corresponding coordinates are equal This implies in particular that they must have the same dimension Philippe B Laval (KSU) Vectors Spring / 18

7 Remarks on Notation Vectors will be denoted with a letter and an arrow on top of the letter, such as u in 2-D or 3-D We will use the following convention If a vector is named u, then we will call u x, u y and u z its x, y, and z-coordinates In other words, in 2-D u = u x, u y and in 3-D u = u x, u y, u z Though your book uses to represent a vector, other texts use parentheses Some texts write vectors in row format, others in column format So, we can have ( ) ux u = u x, u y = (u x, u y ) = The possibilities are the same in 3D u y Philippe B Laval (KSU) Vectors Spring / 18

8 Operations on Vectors: Addition We use the same symbol for vector addition as for addition of numbers We can only add vectors having the same dimension The result is a vector of the same dimension To add two vectors, we simply add their corresponding coordinates For example, in R 2, if u = a, b and v = c, d then u + v = a + c, b + b If we draw the vector v starting at the end of u, then u + v is the vector going from the start of u to the end of v Philippe B Laval (KSU) Vectors Spring / 18

9 Operations on Vectors: Addition Geometrically, vector addition, a + b, can be interpreted two ways The first way involves putting b at the end of a Their sum is the vector starting at the beginning of a and ending at the end of b as shown below Figure: Addition of vectors (triangle law) Philippe B Laval (KSU) Vectors Spring / 18

10 Operations on Vectors: Addition The second way involves drawing both vectors starting at the origin and viewing them as two adjacent sides of a parallelogram The sum is obtained by first completing the parallelogram, then drawing a vector from the origin to the opposite vertex of the parallelogram as hown below Figure: Addition of vectors (parallelogram law) Philippe B Laval (KSU) Vectors Spring / 18

11 Operations on Vectors: Addition Vector addition has the following properties: 1 u + v = v + u (commutative) 2 ( u + v) + w = u + ( v + w) (associative) 3 There exists a vector whose coordinates are all zeros, denoted 0 which satisfies 0 + u = u + 0 for any vector u of the same dimension (existence of an identity element) It is the only vector with this property 4 For any vector u, there exists a unique vector v, whose coordinates are the negative of the coordinates of u, which satisfies u + v = v + u = 0 v is usually denoted u (existence of an additive inverse) Philippe B Laval (KSU) Vectors Spring / 18

12 Operations on Vectors: Scalar Multiplication Scalar multiplication involves multiplying a vector by a number (scalar) It can always be done The result is a vector of the same dimension as the original vector To perform this operation, we simply multiply each coordinate of the vector by the scalar If α is a scalar (number) and u = a, b is a vector, then α u = αa, αb Multiplying a vector by a scalar will stretch the vector ( α > 1) or shrink it (0 < α < 1) If in addition α < 0, the direction of the vector will be reversed We can see that two non-zero vectors are parallel if and only if they are scalar multiples of each other Philippe B Laval (KSU) Vectors Spring / 18

13 Operations on Vectors: Scalar Multiplication Scalar multiplication satisfies the following properties: 1 α ( u + v) = α u + α v for any real number α 2 (α + β) u = α u + β u for any real numbers α and β 3 (αβ) u = α (β u) for any real numbers α and β 4 1 u = u Philippe B Laval (KSU) Vectors Spring / 18

14 Operations on Vectors: Examples Example If u = 1, 1, 1 and v = 1, 2, 3 then find: 1 2 u 2 3 u v Example Ạre u = 1, 1, 1 and v = 1, 2, 3 parallel? Example Ṛepresent graphically u v where u v is defined to be u + ( v) Philippe B Laval (KSU) Vectors Spring / 18

15 Operations on Vectors: Norm of a Vector Definition If u = a, b, the length (or the magnitude, or the norm) of u, denoted u or u, is equal to: u = a 2 + b 2 This is obtained using the distance formula and finding the distance between the origin and the point of coordinates ( a b ) In general, in a space of dimension n, if u = x 1, x 2,, x n then u = x1 2 + x x n 2 Example Find v if v = 1, 2, 3 = n xi 2 i=1 Philippe B Laval (KSU) Vectors Spring / 18

16 Operations on Vectors: Norm of a Vector The norm of a vector satisfies the following properties: 1 u 0 2 u = 0 u = 0 3 α u = α u 4 u + v u + v (triangle inequality) Note the following: 1 Given a non-zero vector u, the vector the same direction as u u u is a unit vector which has 2 Given a non-zero vector u, the vector which has the same direction as u and is of length d, where d > 0, is given by d u u Example Find a vector of length 3 having the same direction as u = 1, 2, 3 Philippe B Laval (KSU) Vectors Spring / 18

17 Standard Basis Vectors 3 vectors play an important role in 3D geometry They are the unit vectors in the direction of the x, y and z-axes They are called i, j, and k They are given by: i = 1, 0, 0 j = 0, 1, 0 k = 0, 0, 1 Any vector u = u x, u y, u z can be represented in terms of the standard basis vectors by u = u x i + u y j + u z k (1) The situation in 2D is similar The standard basis vectors in 2D are i = 1, 0 and j = 0, 1 Example Ẉhat are the components of 3 i + 2 j + k? Philippe B Laval (KSU) Vectors Spring / 18

18 Exercises See the problems at the end of section 12 in my notes on vectors Philippe B Laval (KSU) Vectors Spring / 18