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A vector in the xy-plane may be thought of as an arrow connecting the origin (0,0) to any point (a,b). For example, the following picture shows the vector connecting to the point (10,10) in green and the vector connecting to the point (-3,4) in red.
When v is the vector from (0,0) to (a,b), we write v = (a,b). That is, we identify the arrow with the point at its head. Now why, you may ask, would we go to the trouble of introducing another word to describe points in the plane? Well, the notation v = (a,b) is just a description of the vector that happens to look like a description of a point -- but there is much more involved in the mathematical concept of vector.
First, each such arrow has both a length and a direction. We already know that length and direction are a pair of numbers that can represent a point in the plane -- that's what polar coordinates do. So, in a sense, the vector concept incorporates both rectangular and polar coordinates simultaneously. We use absolute value bars to denote the length of a vector. Thus, if v = (a,b), then .
Second, unlike points in the plane, vectors have an arithmetic. In particular, vectors can be added and subtracted just by adding or subtracting their coordinates:
(a,b) + (c,d) = (a + c, b + d)
(a,b) - (c,d) = (a - c, b - d)
In the next several steps, we will see how to visualize these operations geometrically.
The grey lines in the applet complete a parallelogram with two sides determined by the yellow and green vectors, and the sum is the diagonal of the parallelogram. The grey lines also suggest that it might be useful to free the vectors from the constraint of having their tails tied down at the origin. Specifically, we say that any arrow with the same length and direction as a given arrow from the origin is the same vector as the given one. Thus, the grey side opposite the green arrow is the green arrow (when given the proper direction), and the grey side opposite the yellow arrow is the yellow arrow. When viewed that way, we see that the sum (red arrow) is the third side of a triangle formed when two sides are the yellow and green arrows -- in either order -- with the tail of the second arrow tied to the head of the first.
Third (remember first and second?), a vector can be multiplied by a number ("scaled") to lengthen or shorten it -- and, in the case of a negative multiplier, to reverse its direction. This process is called scalar multiplication, and it is defined by the formula
for a vector v = (a,b) and a scalar c. Since two vectors are parallel if and only if they point in the same or opposite direction, parallel vectors must be scalar multiples of each other. [Note: Numbers are also called scalars because of their role in scaling vectors -- thus, "scalar" is both an adjective and a noun.]
To summarize our "second" and "third" observations about vectors, we have seen that vectors can be combined and manipulated using the vector operations of addition, subtraction, and scalar multiplication.
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