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4-1 Properties of Vectors. 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

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Representing vector quatities
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


Resultant vectors
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


Resultant vectors1
Resultant Vectors

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

    • C

D


Resultant vector
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


Graphical addition of vector
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


Magnitude of the resultant
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


Magnitude of the resultant1
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


Problem
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?



Subtracting vectors
Subtracting Vectors 65

  • 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


Relative velocities
Relative Velocities 65

  • 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


Relative velocities1
Relative Velocities 65

  • 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


Relative velocities in 2 dimensions
Relative Velocities in 2-Dimensions 65

  • 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


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