11 1 11 1 Vectors in the Plane Contemporary Calculus 1 VECTORS IN THE PLANE Measurements of some quantities such as mass speed temperature and height can be given by a single number but a single number is not enough to describe measurements of some quantities in the plane such as displacement or velocity Displacement or velocity not only tell us how much or how fast something has moved but also tell the direction of that movement For quantities that have both length magnitude and direction we use vectors y head 6 A vector is a quantity that has both a magnitude and a direction and vectors are represented geometrically as directed line segments arrows The vector V given by the line segment from the starting point P 2 4 to Q tail V 4 P the point Q 5 6 is shown in Fig 1 The starting point is called the tail of the vector and the ending point is called the head of the vector Geometrically two vectors are equal if they have the same length and point x in the same direction same slope Fig 2 shows a number of vectors that 2 5 Fig 1 are equal to vector V in Fig 1 The equality of vectors in the plane depends only on their lengths and directions Equality of vectors does not y depend on their locations in the plane 6 A vector in the plane can be represented algebraically as an ordered pair of V 4 numbers measuring the horizontal and vertical displacement of the endpoint x of the vector from the beginning point of the vector The numbers in the 2 ordered pair are called the components of the vector Vector V in Fig 1 5 can be represented as V 5 2 6 4 3 2 an ordered pair of numbers enclosed by bent brackets All of the vectors in Fig 2 are also represented algebraically by 3 2 All of these directed line segments represent the vector V Fig 2 Notation In our work with vectors it is important to recognize when we are describing a number or a point or a vector To help keep those distinctions clear we use different notations for numbers points and vectors a number regular lower case letter a b x y x1 y1 A number is called a scalar quantity or simply a scalar a point regular upper case letter A B ordered pair of numbers enclosed by 2 3 a b x1 y1 a vector bold upper case letter A B U V ordered pair of numbers enclosed by 2 3 a b x1 y1 a letter with an arrow over it A B U V 11 1 Vectors in the Plane Definition Contemporary Calculus 2 Equality of Vectors Geometrically two vectors are equal if their lengths are equal and their directions are the same Algebraically two vectors are equal if their respective components are equal if U a b and V x y then U V if and only if a x and b y Example 1 Vectors U and V are given in Fig 3 a Represent U and V using the notation b Sketch U and V as line segments starting at the point 0 0 y c Sketch U and V as line segments starting at the point 1 5 6 4 V d If x y is the starting point what is the ending point of the line segment representing the vector U U x 2 5 Solution a The components of a vector are the displacements from the starting to the ending points so U 2 1 4 2 and V 2 4 6 5 2 1 1 2 Fig 3 y b If U starts at 0 0 then the ending point of the line segment is 0 1 0 2 1 2 V U6 The ending point of V is V 4 0 2 0 1 2 1 See Fig 4 c The ending point of the line segment for U U V U is 1 1 5 2 0 7 For V the ending x 2 5 point is 1 2 5 1 3 6 See Fig 4 d If x y is the starting point for U 1 2 then the ending point is x 1 y 2 Practice 1 A 3 4 and W 2 3 Fig 4 a Represent A and W as line segments beginning at the point 0 0 b Represent A and W as line segments beginning at the point 2 4 c If A and W begin at the point p q at which points do they end d How long is a line segment representing vector A W e What is the slope of a line segment representing vector A W f Find a vector whose line segment representation is perpendicular to A to W 11 1 Vectors in the Plane Contemporary Calculus 3 The magnitude of a vector V written V is the length of the line segment representing the vector That length is the distance between the starting point and the ending point of the line segment The magnitude can be calculated by using the distance formula The magnitude or length of a vector V a b is V a2 b2 The only vector in the plane with magnitude 0 is 0 0 called the zero vector and written 0 or 0 The zero vector is a line segment of length 0 a point and it has no specific direction or slope Adding Vectors If two people are pushing a box in the same direction along a line one with a force of 30 30 pounds pounds and the other with a force of 40 pounds Fig 5 then the result of their efforts is Fig 5 70 pounds 40 pounds equivalent to a single force of 70 pounds along the same line However if the people are pushing in different directions Fig 6 the problem of finding the 40 pounds result of their combined effort is slightly more difficult Vector addition provides a simple solution 30 pounds Fig 6 Definition Vector Addition If A a1 a2 and B b1 b2 then A B a1 b1 a2 b2 The result of applying two forces represented by the vectors A and B is equivalent to the single force represented by the vector A B In Fig 6 the effort of person A can be represented by the vector A 30 0 and the effort of person B by the vector B 0 40 Their combined effort is equivalent to a single force vector C A B 30 40 Since C 50 pounds if the two people cooperated and pushed in the direction of C they could achieve the same result by each exerting 10 pounds less force 11 1 Vectors in the Plane Contemporary Calculus 4 Example 2 Let A 3 5 and B 1 4 a Graph A and B each starting at the origin b Calculate C A B and graph it starting at the origin c Calculate the magnitudes of A B and C d Find a vector V so A V 4 …