6.3 Vectors in the Plane Many quantities in geometry and physics, such as area, time, and temperature, can be represented by a single real number. Other quantities, such as force and velocity, involve both magnitude and direction and cannot be completely characterized by a single real number. To represent such a quantity, we use a directed line segment. The directed line segment PQ has initial point P and terminal point Q and we denote its magnitude (length) by PQ . Q Terminal Point P Initial Point PQ Vector Representation by Directed Line Segments Let u be represented by the directed line segment from P = (0,0) to Q = (3,2), and let v be represented by the directed line segment from R = (1,2) to S = (4,4). Show that u = v. S 4 v 3
Q R 2 u 1 P 1 Using the distance formula, show that u and v have the same length. Show that their slopes are equal. 2 u = 2 (3 0) + (2 0) = 13
v = (4 1)2 + (4 2)2 = 13 2 Slopes of u and v are both 3 2 3 4 Component Form of a Vector The component form of the vector with initial point P = (p1, p2) and terminal point Q = (q1, q2) is PQ = q1 p1 ,q2 p2 = v1 , v2 =v The magnitude (or length) of v is given by 2 2
2 v = (q1 p1 ) + (q2 p2 ) = v1 + v2 2 Find the component form and length of the vector v that has initial point (4,-7) and terminal point (-1,5) 6 Let P = (4, -7) = (p1, p2) and Q = (-1, 5) = (q1, q2). Then, the components of v = v1 , v2 are given by 4 2 -2 -2 2 4
v1 = q1 p1 = -1 4 = -5 v2 = q2 p2 = 5 (-7) = 12 -4 Thus, v = -6 and the length of v is -8 v = (5) 2 + 12 2 = 169 =13 5,12 Vector Operations The two basic operations are scalar multiplication and vector addition. Geometrically, the product of a vector v and a scalar k is the vector that is k times as long as v. If k is positive, then kv has the same direction as v, and if k is negative, then kv has the opposite direction of v. v v
2v -v 3 v 2 Definition of Vector Addition & Scalar Multiplication Let u = u1 ,u 2 and v = v1 , v2 be vectors and let k be a scalar (real number). Then the sum of u and v is u + v = u1 + v1 , u 2 + v2 and scalar multiplication of k times u is the vector ku =k u1 , u2 = ku1 , ku2 Vector Operations Ex. Let v = 2,5 and w = 3,4 . Find the following vectors. a. 2v b. w v 2v= 4,10 w v= 3 (2),4 5 = 5,1
10 2v v 8 4 6 3 4 2 2 -4 -2 -2 w -v
1 2 -1 1 2 3 w-v 4 5 Writing a Linear Combination of Unit Vectors Let u be the vector with initial point (2, -5) and terminal point (-1, 3). Write u as a linear combination of the standard unit vectors of i and j. 6 Solution 10 (-1, 3) 4 8 u = 1 2,3 + 5 2 6
= 3,8 -2 4 u 2 -2 4 8j -4 -6 -8 =3i + 8 j (2, -5) Graphically, it looks like 2 -3i -4 -2 -2 2
Writing a Linear Combination of Unit Vectors Let u be the vector with initial point (2, -5) and terminal point (-1, 3).Write u as a linear combination of the standard unit vectors i and j. Begin by writing the component form of the vector u. u = 1 2,3 (5) u = 3,8 u = 3i + 8 j Unit Vectors v 1 = v u = unit vector = v v Find a unit vector in the direction of v = 2,5
2,5 2 5 v 1 , = = 2,5 = 2 2 v 29 29 29 (2) + (5) Vector Operations Let u = -3i + 8j and let v = 2i - j. Find 2u - 3v. 2u - 3v = 2(-3i + 8j) - 3(2i - j) = -6i + 16j - 6i + 3j = -12i + 19 j