4-1 Properties of Vectors

1. 4-1 Properties of Vectors

2. Representing Vector Quatities • 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

3. Resultant Vectors • 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

4. Resultant Vectors • Vectors shown are unequal, even though they start at the same place • C D

5. Resultant Vector • 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

6. Graphical Addition of Vector • 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

7. Magnitude of the Resultant • 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

8. Magnitude of the Resultant • 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

9. Problem • 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?

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

11. Subtracting Vectors • 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

12. Relative Velocities • 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

13. Relative Velocities • 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

14. Relative Velocities in 2-Dimensions • 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 • v plane relative to air • v plane relative to ground