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Section 6.1b

Section 6.1b. Direction Angles. Velocity, Speed. Let’s start with a brain exercise…. Find the unit vector in the direction of the given vector. Write your answer in (a) component form and (b) as a linear combination of the standard unit vectors i and j. Unit Vector:.

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Section 6.1b

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  1. Section 6.1b Direction Angles Velocity, Speed

  2. Let’s start with a brain exercise… Find the unit vector in the direction of the given vector. Write your answer in (a) component form and (b) as a linear combination of the standard unit vectors i and j. Unit Vector: With standard unit vectors:

  3. Direction Angle – the angle O that a vector makes with the positive x-axis y v |v|sin using trigonometry… x |v|cos Thus, v = (|v|cos 0)i + (|v|sin 0)j And the unit vector in the direction of v is v u = = (cos 0)i + (sin 0)j |v|

  4. Guided Practice Find the components of the vector v with direction angle 123 and magnitude 5. Does this answer make sense graphically ???

  5. Guided Practice Find the magnitude and direction angle of each vector. w = 3, 2

  6. Guided Practice Find the magnitude and direction angle of each vector. w = 5i – 8j

  7. Guided Practice Find the vector v with the given magnitude and the same direction as u. Can we see this problem in a graph? v = 5 u = –5, 7 First, find the unit vector in the direction of u: Now, simply multiply this vector by |v| (the magnitude of v):

  8. Velocity – distance covered per unit time – this is a vector b/c it has both magnitude and direction Speed – the magnitude of velocity (a scalar) Ex: An aircraft is flying on a bearing of 65 at 500mph. Find the component form of the velocity of the plane Start with a graph…do you remember the definition of bearing ?

  9. Ex: An aircraft is flying on a compass heading (bearing) of 350 at 355 mph. A wind is blowing with the bearing 285 at 42 mph. Find (a) the component form of the aircraft’s velocity, and (b) the actual ground speed and direction of the aircraft. (a) (b) Actual speed = 374.688 mph Direction = 344.169 bearing

  10. Cool problem… F F F 2 3 1 Three forces with magnitudes 100, 50, and 80 lb, act on an object at angles of 50 , 160 , and –20 , respectively. Find the direction and magnitude of the resultant force. Start with a diagram: F 1 100 lb F 2 160 50 50 lb –20 80 lb F 3

  11. More of our cool problem… F F F 2 3 1 Three forces with magnitudes 100, 50, and 80 lb, act on an object at angles of 50 , 160 , and –20 , respectively. Find the direction and magnitude of the resultant force. Find the component form of each force: Sum the forces:

  12. Still more for our cool problem… F F F 2 3 1 Three forces with magnitudes 100, 50, and 80 lb, act on an object at angles of 50 , 160 , and –20 , respectively. Find the direction and magnitude of the resultant force. Magnitude of the resultant force: lb Direction of the resultant force: F R 113.808 lb 66.344 lb 92.470 lb

  13. More fun examples!!! A pilot’s flight plan has her flying due east from Flagstaff. There is a 65-mph wind bearing 60 , and the aircraft has a 450 mph speed with no wind. What heading should the pilot follow, and what will be the aircraft’s resultant ground speed? Heading = 94.142 , Speed = 505.116 mph

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