Math 150 – Fall 2015
Section 9A
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Section 9A – Vectors
Definition. A vector is a mathematical object that contains both a direction and a
magnitude. A vector is often represented graphically as an arrow where the direction
is the direction of the arrow, and the magnitude is the length of the arrow.
R2 – Two-dimensional Plane
Definition. In R2 , a vector is represented by an ordered pair ha, bi, and graphically,
the vector is represented by the arrow with the tail at the origin and the tip of the
arrow at the point (a, b).
Example 1. Graph the vectors h3, 4i, h−2, −5i, and h−5, 1i.
Note. A vector is only the direction and length of the arrow, not the starting and
stopping point of the arrow. If two arrows have the same direction and magnitude
(length), but different starting points, then they are still the same vector.
Example 2. Graph the arrow that starts at (2, 1) and stops at (4, 5). Graph the vector
h2, 4i. These vectors have the same direction and magnitude (length) so they are the
same vector.
Definition. In R2 (the plane), there are two special vectors, the vectors of length one in
the positive direction of the x- and y-axis. These are sometimes called the coordinate
vectors. They are denoted by the symbols ~i and ~j, respectively, or ~e1 and ~e2 such that
~i = ~e1 = h1, 0i
~j = ~e2 = h0, 1i
Math 150 – Fall 2015
Section 9A
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R3 – Three dimensional space
Definition. In R3 , a vector is represented by an ordered triple ha, b, ci, and graphically, the vector is represented by the arrow with the tail at the origin and the tip of
the arrow at the point (a, b).
Definition. In R3 (three dimensional space), there are three special vectors, the vectors of length one in the positive direction of the x-, y-, and z-axis. These are sometimes
called the coordinate vectors. They are denoted by the symbols ~i, ~j, and ~k respectively, or ~e1 , ~e2 , and ~e3 such that
~i = ~e1 = h1, 0, 0i
~j = ~e2 = h0, 1, 0i
~k = ~e3 = h0, 0, 1i
Change of Position
Another use of vectors is to show a change of position.
Example 3. Find a vector that which represents moving from the point P (1, 2) to the
point (7, 4). Graph this vector.
Example 4. Find the vector which represents moving from the point P (−5, 2) to the
point Q(3, −3).
Example 5. Find the vector which represents moving from the point P (−4, 2, 1) to
the point (5, 3, −4). Sketch the vector.
Math 150 – Fall 2015
Section 9A
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Example 6. Find the vector which represents moving from the point P (4, −3, −2) to
the point Q(−2, 5, −6). Sketch the vector.
Example 7. A dog runs 560 feet in a direction 43◦ west of north. Assuming that
the origin is the dog’s starting point and north is the positive y-axis, what are the
coordinates of the dog’s location?
Example 8. In the previous problem suppose a flea was on the dog’s back. If after
the dog stops, the flea gets off and goes 1 foot northeast, what are the coordinates of
the flea’s location?
Example 9. If ~x = h−4, −7i, then what angle between 0◦ and 360◦ does ~x make with
the positive x-axis?