VECTOR OPERATION ANALYTICAL GRAPHICAL METHOD GROUP 2: BA L DU EZA , E RW I N CU DA L , A N G E L E M E L INE L I N DE, DI E T HER M E JI A , DE M E TRIO IN AND VECTORS - A vector is an object that has both magnitude and direction. This can be represented by a line segment where the length is the magnitude and the arrow indicates the direction. VECTORS - In vector, we use the terms: ο· Resultant vector – the sum of all vectors ο· Scaling – line transformation that enlarges or diminishes a vector VECTOR OPERATION IN ANALYTICAL METHOD - Analytical method of vector addition and subtraction employs geometry and simple trigonometry. This method is more concise, accurate and precise but, are limited only by the accuracy and precision with which physical quantities are known. - In analytical method, we use the: a. Pythagorean Theorem b. Trigonometric Function c. Component Method 1. A man runs 50m east then 50m north. Determine the resultant vector. a. By Pythagorean Theorem: π 2 = π2 + π 2 π 2 = 502 + (502 ) π2 = C = 70.71 m 502 + (502 ) Vector Addition in Analytical Method 1. A man runs 50m east then 50m north. Determine the resultant vector. b. Solving Ζ, using Trigonometric function. π −1 Ζ = tan ( ) π 50π −1 Ζ = tan ( ) 50π Ζ = πππ πΉ = ππ. πππ, πππ π΅ ππ π¬ (Ans) 2. Two person are pulling a box. The first person applies 6N at NE direction and the second applies 5N at 600 N of W. Scaling: 1N = 1cm 6N = 6cm 5N = 5cm • We can’t solve this problem directly using Trigonometric function and Pythagorean theorem. However, we can use the Component Method where we rewrite the vectors into its components. a. Horizontal component: ο· ππ₯ = π cos Ζ b. Vertical component: ο· ππ¦ = π sin Ζ VECTOR Ζ VX VY 6N 450 + 4.24 + 4.24 5N 600 -2.5 + 4.33 ππ₯ = 1.74 π ππ¦ = 8.57π • To get the resultant, we can now use the Pythagorean theorem: πΆ= ππ₯ + ππ¦ πΆ = 1.74π 2 + 8.57π 2 C = 8.74N • And use the Trigonometric Function to get the Ζ: Ζ= tan−1 ( ππ₯ ) ππ¦ 8.57π Ζ= ) 1.74π Ζ = πππ tan−1 ( R = 8.74N, 780 N of E (Ans.) 1. A man runs 40m west then 40m south. Determine the resultant vector. a. By Pythagorean Theorem: π 2 = π2 + π 2 π 2 = −402 + (−402 ) π2 = C = 56.57 m −402 + (−402 ) Vector Subtraction in Analytical Method 1. A man runs 40m west then 40m south. Determine the resultant vector. b. Solving Ζ, using Trigonometric function. π −1 Ζ = tan ( ) π −40π −1 Ζ = tan ( ) −40π Ζ = πππ πΉ = ππ. πππ ππ πππ πΊπΎ (Ans) VECTOR OPERATION IN GRAPHICAL METHOD - Graphical method of vector in addition and subtraction made use of tools such as rulers and protractors where parts of the vector is retained, because it is represented by arrows for visualization. However, this method is limited by the accuracy with which the drawing or graphs can be made. - In this method, we can use the: a. Head-to-tail or Triangle method – use for 2 given vectors. b. Polygon Method – used for more than 2 given vectors. 1. A man runs 50m east then 50m north. Determine the resultant vector. Scaling: 10m = 1 cm 50 m x 1cm/10m = 5cm ο· Using a ruler, magnitude is 7.1cm Vector Addition in Graphical Method the 10π 7.1 ππ π₯ = πππ 1ππ 5cm ο· Using the protractor, the Ζ is, 450 ο· Thus, the R = 71m at 450 NE. 5cm 2. A bird flies 80m due East, 100m due NE, 110 m at 300 N of W and 160m at 200 S of W. Where is the bird? Scaling: 20m = 1cm 80m = 4cm 100m = 5cm 110m = 5.5cm 160m = 8cm ο· Using the ruler, we calculated the magnitude as 5.9cm 20π 5.9ππ π₯ = ππππ 1ππ ο· Using the protractor, the Ζ is 36.50 ο· Thus, the bird can be found 118m at 36.50 N of W. 1. A man runs 40m west then 40m south. Determine the resultant vector. Scaling: 10m = 1 cm 1ππ 40π π₯ ( ) = 4cm 10π ο· Using a ruler, magnitude is 5.7cm the 10π 5.7 ππ π₯ = πππ 1ππ ο· Using the protractor, the Ζ is, 450 ο· Thus, the R = 57m at 450 SW. Vector Subtraction in Graphical Method
