Chapter 8: Vectors
Geometric Vectors
-A vector is a ________ that has both ___________
and ____________.
Q
-Magnitude:
Notation:
P
-Standard position:
-Direction:
-Zero vector:
O
Mar 15­8:16 AM
Example: Use a ruler and a protractor to determine the
magnitude (in cm) and the direction of n.
n
-Two vectors are _________ if and only if they have the same
___________ and __________.
Let's draw some examples of equal vectors:
Mar 15­9:07 AM
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-Resultant:
Parallelogram Method: Draw the
vectors so that their initial points
coincide. Then draw lines to form a
complete parallelogram. The diagonal
from the initial point to the opposite
vertex of the parallelogram is the
resultant. Note: The parallelogram
method cannot be used to find the sum
of a vector and itself.
Triangle Method: Draw vectors
one after another, placing the
initial point of each successive
vector of the terminal point of
the previous vector. Then draw
the resultant from the initial
point of the first vector to the
terminal point of the last vector.
q
p+q
p+q
q
p
p
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Example: Use both the parallelogram and triangle method to
find the magnitude and direction of v + w.
w
v
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2
-Two vectors are __________ if they have the same magnitude
and opposite directions.
Let's draw some examples of opposite vectors:
-Scalar quantity:
Examples:
-Scalars:
Example: Use your ruler to draw 2w .
w
Mar 15­9:36 AM
Example: Use your ruler and protractor to find the magnitude
and direction of 3u - 2s.
s
u
Mar 15­9:46 AM
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Algebraic Vectors
Vectors can be represented algebraically using ordered pairs of
real numbers.
Example: <3,5> can represent a vector whose initial point is the
origin and whose terminal point is (3,5). It can also represent
any other vector with the same magnitude and direction.
Let P1(x1,y1) be the initial point of a vector and P2(x2,y2) be the
terminal point.
The ordered pair that represents P1P2 is ________________.
The magnitude is given by __________________.
Mar 15­2:30 PM
Example: Write the ordered pair that represents the vector
from X(-3,5) to Y(4,-2). Then find the magnitude of XY.
Vector operations are as expected. So, let's jump to some
examples. Let m = <5,-7>, n = <0,4> and p = <-1,3>. Find each of the
following.
a.) m + p
b.) m - p
c.) 2m + 3n - p
Mar 15­2:48 PM
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-Unit vector:
-A unit vector in the direction of the positive x-axis is
represented by i and a unit vector in the direction of the positive
y-axis is represented by j.
and
j =
i =
Example: Write AB as the sum of unit vectors for A(4,-1) and
B(6,2).
Mar 15­3:00 PM
Vectors in Three-Dimensional Space
Suppose P1(x1,y1,z1) is the initial point of a vector in space and
P2(x2,y2,z2) is the terminal point. The ordered triple that
represents P1P2 is ___________________.
Its magnitude is given by _____________________.
Mar 15­3:11 PM
5
Example: Write the ordered triple that represents the vector
from X(5,-3,2) to Y(4,-5,6). Then find the magnitude.
Example: Find an ordered triple that represents 3p - 2q if
p = <3,0,4> and q = <2,1,-1>.
Mar 15­7:01 PM
Perpendicular vectors
Two vectors are perpendicular if and only if their inner product is
zero.
Let a = <a1,a2> and b = <b1,b2> be vectors.
Inner (Dot) Product of Vectors in a Plane: a b = a1b1 + a2b2
Let a = <a1,a2,a3> and b = <b1,b2,b3>.
Inner (Dot) Product of Vectors in Space: a b = a1b1 + a2b2 + a3b3
Example: Find the inner product of a = <-3,1,1> and b = <2,8,-1>.
Are a and b perpendicular?
Mar 15­7:04 PM
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2123 Chapter 8 Homework
Name: _______________
1.) Use a ruler and protractor to determine the magnitude (in
cm) and direction of the vectors below.
u
v
2.) Use the vectors above to find the magnitude and direction of
each resultant.
a.) u + v
b.) 2u - v
Mar 15­7:35 PM
3.) Given Y(5,0) and Z(7,6). Write the ordered pair that
represents YZ. Then find the magnitude of YZ.
4.) Given b = <6,3> and c = <-4,8>. Find the following:
b.) -4b - ½c
a.) b + c
5.) Find the magnitude of <-3,4>. Then write <-3,4> as the sum of
unit vectors.
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6.) Let T(2,5,4) and M(3,1,-4). Write the ordered triple that
represents TM. Then find the magnitude of TM.
7.) Let v = <4,-3,5>, w = <2,6,-1> and z = <3,0,4>.
Find 3v - (2/3)w + 2z.
Mar 15­7:44 PM
8.) Find each inner product and state whether the vectors are
perpendicular.
c.) <-2,4,8> <16,4,2>
b.) <3,5> <4,-2>
a.) <4,8> <6,-3>
9.) a.) Find two vectors whose magnitude is 10.
b.) Find two vectors that are perpendicular. Prove they are
perpendicular by showing their inner product is 0.
Mar 15­7:50 PM
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