Due at 11:59pm on Friday, August 29, 2008
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To introduce you to vectors and the use of sine and cosine for a triangle when resolving components.
Vectors are an important part of the language of science, mathematics, and engineering. They are used to discuss multivariable calculus, electrical circuits with
oscillating currents, stress and strain in structures and materials, and flows of atmospheres and fluids, and they have many other applications. Resolving a vector
into components is a precursor to computing things with or about a vector quantity. Because position, velocity, acceleration, force, momentum, and angular
momentum are all vector quantities, resolving vectors into components is
the most important skill
required in a mechanics course.
The figure shows the components of
, along the
axes of the coordinate system, respectively. The components of a vector depend on the
coordinate system's orientation, the key being the angle between the vector and the coordinate axes,
The figure shows the standard way of measuring the angle.
axis, and counterclockwise is positive.
in terms of the length of the vector
and the angle
, with the
components separated by a comma.
In principle, you can determine the components of
vector with these expressions. If
lies in one of the other quadrants of the plane,
will be an
angle larger than 90 degrees (or
in radians) and
will have the appropriate signs and values.
Unfortunately this way of representing
, though mathematically correct, leads to equations that must be simplified using trig identities such as
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