Walker, Chapter 3

1
Walker, Chapter 3

• Vectors Trigonometry
• Prof. Charles E. Hyde-Wright
• Physics 111
• Fall 2003

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Vectors and Scalars

• A Scalar is a physical quantity with magnitude
  (and units). Examples
• Temperature, Pressure, Distance, Speed
• A Vector is a physical quantity with magnitude
  and direction
• Displacement Washington D.C. is 250 miles N of
  Norfolk
• Wind Velocity (in a Noreaster) 20mi/hr towards
  SW

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Magnitude and Direction

• Giving directions
• How do I get to the Virginia Beach Boardwalk?
• Go 25 miles. (scalar, almost useless).
• Go 25 miles East. (vector, magnitude direction)

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Components of a vector, as an alternate to
magnitude and direction

• 100 miles 30degrees north of east, is equivalent
  to 86.6 miles east followed by 50 miles north

N
100mi
50mi
86.6mi
E
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Labels for Components of a Vector

• To free ourselves from the points of the compass,
  we will use x y instead of E N
• Vector , magnitude
• Components (as ordered pair)

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Trigonometry and Vector Components

• Trigonometry is not a pre-requisite for this
  course.
• Today you will learn ½ of trigonometry, and all
  that you need for this course.
• In this discussion, we always define the
  direction of a vector in terms of an angle
  counter-clockwise from the x-axis.
• Negative angles are measured clockwise.
• Remember circles, rather than soh-cah-toa

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Trigonometry and Circles

• The point P1(x1,y1) lies on a circle of radius
  r.
• The line from the origin to P1 makes an angle q1
  w.r.t. the x-axis.
• The trigonometric functions sine and cosine are
  defined by the x- and y-components of P1
• x1 r cos(q1) cos(q1) x1 / r
• y1 r sin(q1) sin(q1) y1 / r
• Never mind how to calculate cos sin, let your
  calculator do that.
• Tangent of q1 y1 / x1
• tan(q1) sin(q1) / cos(q1)

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Special (simple) cases of sine and cosine

• cos(0º) 1, sin(0º) 0
• cos(90º) 0, sin(90º) 1
• cos(180º) -1, sin(180º) 0
• cos(270º) 0, sin(270º) -1
• Sine and Cosine are periodic functions
• cos(q360) cos(q)
• sin(q360º) sin(q)

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45-45-90 triangle

• By symmetry,
• x1 y1
• Pythagorous
• x12 y12 r2
• 2 x12 r2
• x1 r/?2
• cos(45º) x1 /r 1/?2
• cos(45º) 0.7071
• Sin(45º) 1/ ?2

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30-60-90 Triangle

• From Equilateral triangle
• 2y1 r
• Pythagorous

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Navigating the Quadrants(Circles are better than
Triangles)

• First Quadrant
• 0º lt q lt 90º
• cos(q) gt 0, sin(q) gt 0
• Second Quadrant
• 90º lt q lt 180º
• cos(q) lt 0, sin(q) gt 0
• Third Quadrant
• 180º lt q lt 270º
• cos(q) lt 0, sin(q) lt 0
• Forth Quadrant
• 0º lt q lt 90º
• cos(q) gt 0, sin(q) lt 0

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Moving from Quadrant to QuadrantAdding 180
degrees

• q2q1180?
• x2 -x1, y2 -y1
• cos(q1 180º) -cos(q1)
• sin(q1 180º) -sin(q1).

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Moving from Quadrant to QuadrantSupplementary
angles (reflection about y-axis)

• q2 180? - q1
• x2 -x1, y2 y1
• cos(q2) x2 /r
• sin(q2) y2 /r
• cos(180º -q1) -cos(q1)
• sin(180º -q1) sin(q1).

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Inverting the sign of an angle(reflection about
x-axis)

• q2 - q1
• x2 x1, y2 -y1
• cos(q2) x2 /r
• sin(q2) y2 /r
• cos(-q1) cos(q1)
• Cosine is an EVEN function
• sin(-q1) -sin(q1).
• Sine is an ODD function

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Complementary Angles

• q2 90? - q1
• x2 y1, y2 x1
• cos(q2) x2 /r y1 /r
• sin(q2) y2 /r x2 /r
• cos(90º -q1) sin(q1)
• sin(90º -q1) cos(q1).
• Valid for any value of q1.

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Vector AdditionGraphical(use bold face for
vector symbol)

• A, B, and C are three displacement vectors.
• Any point can be the origin for a displacement
• The vector B 3 paces to E.
• Notice that B has been translated from the origin
  until the tail of B is at the head of A.
• This is the head-to-tail method of vector
  addition.
• Vector addition is commutative, just like
  ordinary addition
• D ABC CBA

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Vector Addition, Components

• When we add two vectors, the components add
  separately
• Cx Ax Bx Bx Ax
• Cy Ay By By Ay

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Velocity Vectors

• Each fish in a school has its own velocity
  vector.
• If the fish are swimming in unison, the velocity
  vectors are all (nearly) identical
• We draw each vector at the position of the fish.

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Scalar MultiplicationMultiplying a vector by a
scalar

• Multiplying a vector by a positive scalar
  quantity simply re-scales the length (and maybe
  units) of the vector, without changing direction.
• Multiplying a vector by a negative number
  reverses the direction of the vector.

y
x
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Vector Subtraction

• Subtraction is just addition of the additive
  inverse

y
x
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Average Velocity Vector

• Net displacement (vector) multiplied by
  reciprocal of elapsed time (scalar)

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Relative Motion
vpg -1.2m/s15m/s 13.8m/s
vpg 1.2m/s15m/s16.2m/s
vpg velocity of person (hobo) relative to
ground
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Quiz 1 Row-Row-Row your Boat

• You are rowing across a river, with the boat
  pointed 25 degrees upstream.
• Does the time it takes you to cross the river
  depend upon the speed of the river?
• A Yes B No C Depends on current
• Hint think about velocity components
  perpendicular and parallel to river direction.
• Does the current in the river change the velocity
  component of the boat, perpendicular to
  riverbank?
• Hint The vector at right is the velocity vector
  of the boat relative to the water. Draw the
  velocity vector of the water, and the velocity
  vector of the boat relative to shore.

25º