Vectors. Vector Multiplication

Transcription

1 Vectors Directed Line Segments and Geometric Vectors A line segment to which a direction has been assigned is called a directed line segment. The figure below shows a directed line segment form P to Q. We call P the initial point and Q the terminal point. We denote this directed line segment by PQ. Q Initial point P The magnitude of the directed line segment PQ is its length. We denote this by PQ. Thus, PQ is the distance from point P to point Q. Because distance is nonnegatie, ectors do not hae negatie magnitudes. Geometrically, a ector is a directed line segment. Vectors are often denoted by a boldface letter, such as. If a ector has the same magnitude and the same direction as the directed line segment PQ, we write = PQ. Terminal point Vector Multiplication If k is a real number and a ector, the ector k is called a scalar multiple of the ector. The magnitude and direction of k are gien as follows: The ector k has a magnitude of k. We describe this as the absolute alue of k times the magnitude of ector. The ector k has a direction that is: the same as the direction of if k > 0, and opposite the direction of if k < 0

2 The Geometric Method for Adding Two Vectors A geometric method for adding two ectors is shown below. The sum of u + is called the resultant ector. Here is how we find this ector.. Position u and so the terminal point of u extends from the initial point of. 2. The resultant ector, u +, extends from the initial point of u to the terminal point of. Resultant ector u + Terminal point of u Initial point of u The Geometric Method for the Difference of Two Vectors The difference of two ectors, u, is defined as u = + (-u), where u is the scalar multiplication of u and : -u. The difference u is shown below geometrically. -u u -u u The i and Unit Vectors Vector i is the unit ector whose direction is along the positie x-axis. Vector is the unit ector whose direction is along the positie y-axis. y O i x 2

3 Representing Vectors in Rectangular Coordinates Vector, from (0, 0) to (a, b), is represented as = ai + b. The real numbers a and b are called the scalar components of. Note that a is the horizontal component of, and b is the ertical component of. The ector sum ai + b is called a linear combination of the ectors i and. The magnitude of = ai + b is gien by = a 2 + b 2 Text Example Sketch the ector = -3i + 4 and find its magnitude. Solution For the gien ector = -3i + 4, a = -3 and b = 4. The ector, shown below, has the origin, (0, 0), for its initial point and (a, b) = (-3, 4) for its terminal point. We sketch the ector by drawing an arrow from (0, 0) to (-3, 4). We determine the magnitude of the ector by using the distance formula. Thus, the magnitude is = a 2 + b 2 = ( 3) = 9+ 6 = 25 = 5 Terminal point = -3i Initial point -4-5 Representing Vectors in Rectangular Coordinates Vector with initial point P = (x, y ) and terminal point P 2 = (x 2, y 2 ) is equal to the position ector = (x 2 x )i + (y 2 y ). 3

4 Adding and Subtracting Vectors in Terms of i and If = a i + b and w = a 2 i + b 2, then + w = (a + a 2 )i + (b + b 2 ) w = (a a 2 )i + (b b 2 ) If = 5i + 4 and w = 6i 9, find: a. + w b. w. Solution Text Example + w = (5i + 4) + (6i 9) These are the gien ectors. = (5 + 6)i + [4 + (-9)] Add the horizontal components. Add the ertical components. = i 5 Simplify. + w = (5i + 4) (6i 9) These are the gien ectors. = (5 6)i + [4 (-9)] Subtract the horizontal components. Subtract the ertical components. = -i + 3 Simplify. Scalar Multiplication with a Vector in Terms of i and If = ai + b and k is a real number, then the scalar multiplication of the ector and the scalar k is k = (ka)i + (kb). 4

5 Example If =2i-3, find 5 and -3 Solution: 5= (5*2) i+ (5* 3) = 0i 5 3= ( 3*2) i+ ( 3* 3) = 6i+ 9 The Zero Vector The ector whose magnitude is 0 is called the zero ector, 0. The zero ector is assigned no direction. It can be expressed in terms of i and using 0 = 0i + 0. Properties of Vector Addition and Scalar Multiplication If u,, and w are ectors, then the following properties are true. Vector Addition Properties. u + = + u Commutatie Property 2. (u + ) + w = + (u + w)associatie Property 3. u + 0 = 0 + u = u Additie Identity 4. u + (-u) = (-u) + u = 0 Additie Inerse 5

6 Properties of Vector Addition and Scalar Multiplication If u,, and w are ectors, and c and d are scalars, then the following properties are true. Scalar Multiplication Properties. (cd)u = c(du) Associatie Property 2. c(u + ) = c + cu Distributie Property 3. (c + d)u = cu + du Distributie Property 4. u = u Multiplicatie Identity 5. 0u = 0 Multiplication Property 6. c = c Finding the Unit Vector that Has the Same Direction as a Gien Nonzero Vector For any nonzero ector, the ector is a unit ector that has the same direction as. To find this ector, diide by its magnitude. Example Find a unit ector in the same direction as =4i-7 Solution: = = ( 7) 6+ 49= = 4 i

7 Vectors 7