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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.