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4-1 Properties of Vectors

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4-1 Properties of Vectors

- The line and arrow used in Ch.3 showing magnitude and direction is known as the Graphical Representation
- Used when drawing vector diagrams

- When using printed materials, it is known as Algebraic Representation
- Italicized letter in boldface
- d = 50 km SW

- Two displacements are equal when the two distances and directions are equal
- A and B are equal, even though they don’t begin or end at the same place
This property of vectors makes it possible to move

vectors graphically for adding or subtracting

- A and B are equal, even though they don’t begin or end at the same place

A

- Vectors shown are unequal, even though they start at the same place
- C

D

- The resultant vector is the displacement of the vector additions.
- My route to school is
- My resultant vector is R
- 0.50 miles East
- 2.0 miles North
- 2.5 miles East
- 20.0 miles North
- 2.5 miles East
- Resultant Vector = 23 miles NE

R

- When manipulating graphical reps. of vectors, need a ruler to measure correct length
- Take the tail end and place at the head of the arrow
- Enroute to a school, someone travels 1.0 km W, 2.0 km S, and then 3.0 km W
- Resultant vector =
- 4.5 km SW

- Vectors added at right angels can use the Pythagorean System to find magnitude
- If vectors added and angle is something other than 90o, use the Law of Cosines
- R2 = A2 + B2 – 2ABcosθ

- Find the magnitude of the sum of a 15 km displacement and a 25 km displacement when the angle between them is 135o.
- A = 15 km; B = 25 km; θ = 135o; R = unknown
- R2 = A2 + B2 – 2ABcosθ
- = (25 km)2 + (15 km)2 – 2(25km)(15 km)cos135o
- =625 km2 + 225 km2 – 750km2(-0.707)
- =1380 km2
- R = √1380km2
- = 37 km

- A hiker walks 4.5 km in one direction, then makes a 45o turn to the right and walks another 6.4 km. What is the magnitude of her displacement?

- A person walked 450.0 m North. The person then turned left 65o and traveled 250.0 meters. Find the resultant vector.

- Multiplying a Vector by a scalar number changes its length, but not direction, unless negative
- Vector direction is then reversed
- To subtract two vectors, reverse direction of the 2nd vector then add them
- Δv = v2 – v1
- Δv = v2 + (-v1)
- If v1 is multiplied by -1, the direction of v1 is reversed and can be added to v2 to get Δv

- Graphical addition can be used when solving problems that involve relative velocity
- School bus traveling at a velocity of 8 m/s. You walk toward the front at 3 m/s. How fast are you moving relative to the street?
- vbusrelative to street
- vyourelative to bus
- vyourelative to the street

- When a coordinate system is moving, two velocities add if both moving in the same direction & subtract if the motions are in opposite directions
- What if you use the same velocities and walk to the rear of the bus. What is your resultant velocity relative to the street?
- vbus relative to the street
- vyou relative to the bus
- vyou relative to the street

- Suppose an airplane pilot wants to fly from the U.S. to Europe. Does the pilot aim the plane straight to Europe?
- No, must take in consideration for wind velocity
- v air relative to the ground

- No, must take in consideration for wind velocity

- v plane relative to ground