4 1 properties of vectors
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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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4-1 Properties of Vectors

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4 1 properties of vectors

4-1 Properties of Vectors


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?


4 1 properties of vectors

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


Subtracting vectors

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


Relative velocities

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


Relative velocities1

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


Relative velocities in 2 dimensions

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


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