644 CHAPTER 9 Polar Coordinates; Vectors magnitude and direction, rather than in terms of its components. For example, a ball thrown with an initial speed of 25 miles per hour at an angle of ° 30 to the horizontal is a velocity vector. Suppose that we are given the magnitude v of a nonzero vector v and the direction angle α α ° ≤ < ° , 0 360 , between v and i. To express v in terms of v and α, first find the unit vector u having the same direction as v . Look at Figure 61. The coordinates of the terminal point of u are α α ( ) cos , sin . Then α α = + u i j cos sin and, from equation (7), Figure 61 v v i j cos sin α α ( ) = + 1 1 y x a i j (cos a, sin a) u v Figure 62 y x 12.5 j 21.65 i 12.5 21.65 v = 25(cos 308i + sin 308j) 25 308 Figure 63 x a v 5 4i 2 4j (4,24) 24 4 Figure 64 Resultant force F1 1 F2 Resultant F1 F2 α α ( ) = + v v i j cos sin (8) where α is the direction angle between v and i . Finding a Vector When Its Magnitude and Direction Are Given A ball is thrown with an initial speed of 25 miles per hour in a direction that makes an angle of ° 30 with the positive x -axis. Express the velocity vector v in terms of i and j . What is the initial speed in the horizontal direction? What is the initial speed in the vertical direction? EXAMPLE 6 Solution The magnitude of v is = v 25 miles per hour, and the angle between the direction of v and i , the positive x -axis, is α = ° 30 . By equation (8), α α ( ) ( ) = + = ° + ° v v i j i j cos sin 25 cos30 sin 30 = + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = + i j i j 25 3 2 1 2 25 3 2 25 2 The initial speed of the ball in the horizontal direction is the horizontal component of v , ≈ 25 3 2 21.65 miles per hour. The initial speed in the vertical direction is the vertical component of v , = 25 2 12.5 miles per hour. See Figure 62. Now Work PROBLEM 61 Finding the Direction Angle of a Vector Find the direction angle α of = − v i j 4 4 . Solution EXAMPLE 7 Figure 63 shows the vector v and its direction angle α. To find α, use the terminal point ( ) − 4, 4 and the fact that α = − = − tan 4 4 1 Because α ° ≤ < ° 0 360 , the direction angle is α = ° 315 . CAUTION If you use a calculator to find ( ) − − tan 1 , 1 the calculator will only show values between − ° 90 and ° 90 . j Now Work PROBLEM 67 7 Model with Vectors Because forces can be represented by vectors, two forces “combine” the way that vectors “add.” If F1 and F2 are two forces simultaneously acting on an object, the vector sum + F F 1 2 is the resultant force .The resultant force produces the same effect on the object as that obtained when the two forces F1 and F2 act on the object, one after the other. See Figure 64.
RkJQdWJsaXNoZXIy NjM5ODQ=