1. Vectors Topic 1 (Cont.)

5. Addition of Vectors — Graphical Methods For vectors in one dimension, simple addition and subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates.

6. If the motion is in two dimensions, the situation is somewhat more complicated. Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem.

7. Adding the vectors in the opposite order gives the same result:

8. Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.

9. The parallelogram method may also be used; here again the vectors must be tail-to-tip.

10. Subtraction of Vectors In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction. Then we add the negative vector.

11. Multiplication of a Vector by a Scalar A vector can be multiplied by a scalar c ; the result is a vector c that has the same direction but a magnitude cV . If c is negative, the resultant vector points in the opposite direction.

12. If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B ? 1) same magnitude, but can be in any direction 2) same magnitude, but must be in the same direction 3) different magnitudes, but must be in the same direction 4) same magnitude, but must be in opposite directions 5) different magnitudes, but must be in opposite directions ConcepTest 3.1a Vectors I

13. If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B ? 1) same magnitude, but can be in any direction 2) same magnitude, but must be in the same direction 3) different magnitudes, but must be in the same direction 4) same magnitude, but must be in opposite directions 5) different magnitudes, but must be in opposite directions The magnitudes must be the same, but one vector must be pointing in the opposite direction of the other in order for the sum to come out to zero. You can prove this with the tip-to-tail method. ConcepTest 3.1a Vectors I

14. Given that A + B = C , and that l A l 2 + l B l 2 = l C l 2 , how are vectors A and B oriented with respect to each other? 1) they are perpendicular to each other 2) they are parallel and in the same direction 3) they are parallel but in the opposite direction 4) they are at 45 ° to each other 5) they can be at any angle to each other ConcepTest 3.1b Vectors II

15. Given that A + B = C , and that l A l 2 + l B l 2 = l C l 2 , how are vectors A and B oriented with respect to each other? 1) they are perpendicular to each other 2) they are parallel and in the same direction 3) they are parallel but in the opposite direction 4) they are at 45 ° to each other 5) they can be at any angle to each other Note that the magnitudes of the vectors satisfy the Pythagorean Theorem. This suggests that they form a right triangle, with vector C as the hypotenuse. Thus, A and B are the legs of the right triangle and are therefore perpendicular. ConcepTest 3.1b Vectors II

16. Given that A + B = C , and that l A l + l B l = l C l , how are vectors A and B oriented with respect to each other? 1) they are perpendicular to each other 2) they are parallel and in the same direction 3) they are parallel but in the opposite direction 4) they are at 45 ° to each other 5) they can be at any angle to each other ConcepTest 3.1c Vectors III

17. Given that A + B = C , and that l A l + l B l = l C l , how are vectors A and B oriented with respect to each other? 1) they are perpendicular to each other 2) they are parallel and in the same direction 3) they are parallel but in the opposite direction 4) they are at 45 ° to each other 5) they can be at any angle to each other The only time vector magnitudes will simply add together is when the direction does not have to be taken into account (i.e., the direction is the same for both vectors). In that case, there is no angle between them to worry about, so vectors A and B must be pointing in the same direction. ConcepTest 3.1c Vectors III

18. Adding Vectors by Components Any vector can be expressed as the sum of two other vectors, which are called its components. The process of finding the component is known as resolving the vector into its component.

19. Because x and y axis is perpendicular, they can be calculate using trigonometric functions.

20. The components are effectively one-dimensional, so they can be added arithmetically.

22. Example 3-2: Mail carrier’s displacement. A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0° south of east for 47.0 km. What is her displacement from the post office?

24. Example 3-3: Three short trips. An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast (45°) for 440 km; and the third leg is at 53° south of west, for 550 km, as shown. What is the plane’s total displacement?

25. If each component of a vector is doubled, what happens to the angle of that vector? 1) it doubles 2) it increases, but by less than double 3) it does not change 4) it is reduced by half 5) it decreases, but not as much as half ConcepTest 3.2 Vector Components I

26. If each component of a vector is doubled, what happens to the angle of that vector? 1) it doubles 2) it increases, but by less than double 3) it does not change 4) it is reduced by half 5) it decreases, but not as much as half The magnitude of the vector clearly doubles if each of its components is doubled. But the angle of the vector is given by tan  = 2y/2x , which is the same as tan  = y/x (the original angle). Follow-up: If you double one component and not the other, how would the angle change? ConcepTest 3.2 Vector Components I