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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 A This property of vectors makes it possible to move vectors graphically for adding or subtracting 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? ◦ vbus relative to street ◦ vyou relative to bus ◦ vyou relative 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 v air relative to the ground plane relative to air v plane relative to ground