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Thinking Ahead about Vectors II – You should understand these concepts fully before the next class. Check
your answers with the key on your instructor’s website. You can get help with this work from the following
sources:
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Visit your instructor during office hours
Go to the MAC (Math Assistance Center) 700 BH
Review Thinking Ahead about Vectors I
Go to http://zonalandeducation.com/mstm/physics/mechanics/vectors/vectors.html
Or https://www.khanacademy.org/science/physics/one-dimensional-motion/displacement-velocitytime/v/introduction-to-vectors-and-scalars
Or http://www.nhn.ou.edu/walkup/demonstrations/WebTutorials/VectorIntroduction.htm
Or http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-introvector-2009-1.pdf
Vector quantities may be represented graphically, as shown in Thinking Ahead about Vectors I, but also may
be specified numerically, which allows a more precise description than graphical representations.
To specify a two-dimensional vector quantity, two numbers are required. (For three dimensions, three numbers
are required.) The two numbers may be a magnitude (labeled r in this case) and a direction angle (labeled  in
this case) measured from a reference line, such as the x-axis. Alternatively, using rectangular coordinates, the
two numbers may be two magnitudes, one measured parallel to the x-axis, and one measured parallel to the yaxis. These two numbers would suffice to specify the vector. These represent the rectangular components, x
and y, of the vector.
y
r
y
x
x
As an example, if x = 4 meters and y = 3 meters, we could state that the vector displacement, r, is 4 meters
(parallel to the x-axis) + 3 meters (parallel to the y-axis). The magnitude of the displacement would be
r  42  32  5 meters, and the direction angle   arctan  3 4   36.9 .
Instead of writing out or saying “parallel to the x-axis” (or y-axis), quantities called unit vectors are normally
used. Each unit vector is parallel to a particular coordinate axis (direction), has a magnitude of 1, and no
dimensions. The common notation is iˆ for a unit vector parallel to the x-axis, ĵ parallel to y, and k̂ parallel to z.


Then to specify the displacement vector given above, r  4iˆ  3 ˆj meters, which means exactly the same thing
as 4 meters (parallel to the x-axis) + 3 meters (parallel to the y-axis).


Given a vector in unit-vector notation, such as, r  5iˆ  12 ˆj meters, it is straightforward to find the magnitude
and direction of the vector. By Pythagoras’ theorem, the magnitude is r  52  122  13 meters. The direction
is given by trigonometry as   arctan 12 5  67.4 measured counterclockwise from a reference line parallel
to the positive x-axis.
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Given the magnitude and direction of a vector, such as r = 10 meters at an angle of 30° measured
counterclockwise from a reference line parallel to the positive x-axis, it is straightforward to find the unit-vector
representation. r  10cos30 iˆ  10sin 30 ˆj  8.66iˆ  5 ˆj meters.
y
10
30° 10sin30°
10cos30°
x
Given two or more vectors in unit-vector form, addition or subtraction of the vectors is very easy. For instance
if r1  4iˆ  3 ˆj meters and r2  5iˆ  12 ˆj meters, then r1  r2   5  4  iˆ   3  12  ˆj  9iˆ  16 ˆj meters. The sum


has a magnitude of


92  162  18.4 meters and a direction   arctan 16 9   60.6 .
Also, r1  r2   5  4  iˆ   3  12  ˆj  iˆ  9 ˆj meters. This vector is in the fourth quadrant of the rectangular
coordinate system, and has a magnitude of 12   9   9.06 meters and a direction   arctan  9 1  83.7 ,
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or 83.7° measured clockwise from the positive x-direction.
Given more vectors, follow the same procedure, and add each of the components separately –
iˆ to iˆ ˆj to ˆj and kˆ to kˆ. Do not mix iˆ with ˆj or kˆ components.
Exercises: Given r1  3iˆ  4 ˆj  2kˆ, r2  5iˆ  2 ˆj  3kˆ, r3  6iˆ  4 ˆj  3kˆ find the following in unit vector form:
a) r1  r2  r3
b) r1  r2  r3
c) r1  r2  r3
d) r3  2r2  r1 (hint: 2r2 is a vector with each component twice as large as the components of r2 . )
e) r2  r3
f) Find the magnitude and direction of the vector of part e). State the direction as an angle measured
counterclockwise from the positive x-direction. Be careful to note the quadrant of the result, or the direction
angle will be incorrect.