Download 12.8 Algebraic Vectors and Parametric Equations Algebraic Vectors

12.8 Algebraic Vectors and Parametric Equations
Algebraic Vectors
• Every vector can be translated so that its tail is at
the ____________ and head is at
v
______________. Denoted:____________.
• Coordinates represent ____________________.
• The magnitude of v is:
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Polar Coordinates
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Unit Vectors
• Has magnitude equal to _________.
• Unit vector in horizontal direction is denoted _____________.
• Unit vector in vertical direction is denoted _____________.
• All vectors can be expressed as _______________________.
> Example: (3, 4)
> Component form:
> Polar form:
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• If v is any nonzero vector, then ______ is the unit vector in the same
direction as v.
> Example: Find the unit vector in the same direction as v if v = (3,4)
• A vector can be determined in component or polar form when the
vector is described by its ___________ and ___________ points.
The vector can be described by its _________________ and
_____________ components as follows:
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Example 1: Given v with an initial point of (2,3) and a terminal point of
(7,9), determine the:
a.) component form
b.) unit vector in the same direction
c.) polar form
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Vector Operations (Algebraically)
Given vectors v = (x,y) and w = (r,s), then:
Vector Sum:
Vector Difference:
Scalar Multiplication:
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Example 2: Given s = (3,5), w = (-6,1), and u = (7, -3), determine a vector v
that satisfies the given equation.
a.) v = s+w
b.) v = u - w
c.) v = 2s + u
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A geometric representation of 2c:
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Generally:
Vector Equation:
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Example 3:
a.) Find a vector equation of the line RS through R(1,5) with direction
vector v=(2,3).
b.) Find points on RS other than R(1,5).
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When given two points on a line, a direction vector can be determined if:
Suppose (c,d) is a direction vector of PQ. When the vector equation is
simplified by scalar multiplication and vector addition, the equation can be
written:
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Example 4: Determine a direction vector of the line containing the two
points P(5, 8) and Q(11, 2). Then find a vector equation of the line, a pair of
parametric equations of the line, and 3 additional points on PQ.
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Example 5: Parametric equations can be used to model real world
phenomena, especially when more than one variable is dependent upon time.
If an object is launched into the air at an angel of θ degrees with the
ground and an initial velocity of magnitude v0 ft/s, then the position of the
object may be described by the parametric equations:
A space vehicle rises from its launch pad and reaches a height of 5 mi.
Because of a computer malfunction, the booster rockets are turned off
prematurely, when the vehicle is moving at a velocity of 6 mi/s at an angle
of 40 degrees with an imaginary reference plane tangent to the earth
directly below. The rocket's velocity is thus below the velocity neeed to
attain earth orbit. Determine: a) parametric equations to model its flight;
b) number of seconds for it to hit the ocean; and c) horizontal distance
traveled by the vehicle.
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