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The displacement of a man walking through the desert can be represented
by a vector: the magnitude of the vector corresponds to the distance he
has walked, while the direction of the vector corresponds to the
direction in which he has walked. (This is perhaps the most intuitive example
of vectors being applied to real problems). The next 5 questions will deal with
such displacement vectors.
The velocity of a moving car can be represented by a vector: the
magnitude of the vector corresponds to the speed of the car, the
direction of the vector corresponds to the direction in which the car is
moving. If the car speeds up or slows down, the magnitude of its corresponding
velocity vector changes (gets longer or shorter). If the car turns, the
direction of its velocity vector is also altered (and will rotate to point in
the new direction in which the car is heading). This idea will be useful in
answering the following 5 questions.
A line passing through the origin in 3-dimensional space can be characterized by
a vector v: all points on the line can be written in the form
tv, where t is a real number (t = 0 yields the origin itself), and
the full set of points (ranging over all values of t) is the whole line. For a
line that does not pass through the origin, all the points can be written of the
form u + tv, where u is a particular vector which can be
chosen at will from the points which lie on the line. This idea is central to
the following 5 questions.