1. Unit 37 VECTORS

2. DEFINITIONS A vector is a quantity that has both magnitude and direction Vectors are shown as directed line segments. The length of the segment represents the magnitude and the arrowhead represents the direction of the quantity Vectors have an initial point and a terminal point. An arrowhead represents the terminal point A vector is named by its two end points or by a single lowercase letter. Arrows are placed above the line segments 2

3. DEFINITIONS Equal vectors have identical magnitudes and directions. A vector can be repositioned provided its magnitude and direction remain the same Vectors are usually shown on the rectangular coordinate (x,y) system. A vector is in standard position when the initial point is at the origin of the rectangular coordinate (x,y) system and its angle is measured counterclockwise from the positive x-axis 3

4. EXAMPLES The vector shown below could be named either Vectors below are not equal vectors because they do not have identical magnitudes and directions. 4 A B f a b

5. VECTOR NOTATION A vector can be represented on the rectangular coordinate system using ordered pair notation (Ch 15) (x,y), or vectors can be shown and solved as lengths and angles. Note: Vector angles are often represented as  (theta) The vector shown on the following slide could either be read as 3x + 4y [(3,4)cm] in the rectangular coordinate system or as 5cm at 53.1° (53.1 and 5 is found using trig/Pythagorean Thrm…make a right triangle) 5

6. VECTOR NOTATION 6

7. ADDING VECTORS A resultant vector is the sum of two or more vectors. Component vectors are the vectors that are added to produce the resultant vector The resultant vector produces the same effect as the combination of two or more component vectors. The resultant vector can be substituted for the component vectors or the component vectors can be substituted for the resultant vector There are two basic methods of adding vectors: the graphic method and the trigonometric method. The trigonometric method allows for virtually any degree of accuracy 7

8. ADDING VECTORS GRAPHICALLY Vectors (3,4) and (4,5) were added graphically below by completing the following steps: 8 1. First vector (3,4) was drawn in standard position 2. Second, vector (4,5) was drawn with its initial point on the terminal point of vector (3,4) 3. Finally, the resultant vector was drawn from the origin to the terminal point of vector (4,5) and its length and angle were measured to give us the resultant vector of 11.4 at 52.1° bb (4,5) aa (3,4) x y RR

9. ADDING VECTORS USING TRIGONOMETRY The steps listed below give the general (component vector) procedure for adding vectors using trigonometry: 9 1. Compute the horizontal (x) component of each vector. Add all the x-components algebraically. The sum is the x-component of the resultant vector 2. Compute the vertical (y) component of each vector. Add all the y- components algebraically. The sum is the y-component of the resultant vector 3. Compute the magnitude of the resultant vector using the Pythagorean Theorem 4. Compute the reference angle using the tangent function

10. TRIGONOMETRY EXAMPLE Note: 120° would have a reference angle of 60° in the second quadrant 10 140 lb 45° 65 lb 60° xaxa yaya xbxb ybyb First, determine the x-components: Next, determine the y-components: (Note: x b would be –70 because it is in quadrant II)

11. TRIGONOMETRY EXAMPLE 11 Now, add the components determined on the previous slide: Thus, the resultant vector is (–24.04, 167.2) lbs. 24.04 167.2 R  Determine the magnitude (R): Find the angle: so  = –81.82°. But, since the resultant is in quadrant II;  = 180° – 81.82° = 98.18° So, R = 168.92 lbs. at 98.18° Ans x a + x b = 45.96 lbs. + –70 lbs. = –24.04 lbs. y a + y b = 45.96 lbs. + 121.24 lbs. = 167.2 lbs.

12. PRACTICE PROBLEMS 1. Which of the following quantities are vectors? a. 47 pounds per square inch b. 65 miles westward c. 75 miles per hour 2. Which of the following show or name vectors? a. b. c. d. e. AE 3. Add vector (4,5) inches and vector (–2,3) inches graphically. 4. Determine the vector sum of a force of 1240 pounds vertically up and another force of 1510 pounds horizontally to the right. 12

13. PRACTICE PROBLEMS (Cont) 5. Compute the resultant vector for the following sets of vectors using trigonometry. a = 200 lbs. at 47° b = 150 lbs. at 98.6° c = 260 lbs. at 187.5° 6. Compute the resultant vector for the following sets of vectors using trigonometry. d = 1500 miles at 230° e = 1430 miles at 154.4° f = 1375 miles at 345° 13

14. PROBLEM ANSWER KEY 1. b 2. b, c, and e 3. 8.25 inches at 76° (See graph at right) 14

15. PROBLEM ANSWER KEY (Cont) 4. 1953.89 lbs. at 39.4° (See graph at right) 5. 297.69 lbs. at 118.9° 6. 1282.07 miles at –136.2° (or 223.8°) 15