A vector is represented in the coordinate plane by a directed line segment whose length gives its magnitude and whose orientation gives its direction; translating the segment without changing either preserves the same vector. Vectors may be described by initial and terminal points or by horizontal and vertical components, with geometric addition modeled by placing one directed segment after another or using the parallelogram rule. This treatment excludes abstract vector spaces and higher-dimensional generalizations.
A vector in the coordinate plane is shown by a directed line segment. The arrow’s length represents the vector’s magnitude, and the arrow’s direction represents its orientation. Moving the arrow to a different location without rotating or stretching it gives the same vector.
Example: Represent the vector from to geometrically.
Plot the two points.
Place and on the coordinate plane.
Draw the directed segment.
Draw a line segment from to and place the arrowhead at . The vector is written as
The arrow points from the initial point to the terminal point .
Find the horizontal and vertical changes.
From to :
Therefore, the vector’s components are
Translate the vector to the origin.
To draw an equivalent vector with its initial point at , move the arrow units right and units up. Its terminal point will be .
Thus, the directed segment from to represents the same vector:
The two arrows have the same length and point in the same direction, even though they start at different locations. Therefore, they represent the same vector.
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