6.3 Vectors in the Plane Many quantities in

6.3 Vectors in the Plane Many quantities in

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

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