A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector.

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A vector is a quantity that has both magnitude
and direction. It is represented by an arrow.
The length of the vector represents the magnitude
and the arrow indicates the direction of the
vector.
Blue and orange vectors have same magnitude but
different direction.
Blue and green vectors have same direction but
different magnitude.
Blue and purple vectors have same magnitude and
direction so they are equal.
Two vectors are equal if they have the same
direction and magnitude (length).
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Q
How can we find the magnitude if we have the
initial point and the terminal point?
The distance formula
Terminal Point
magnitude is the length
direction is this angle
Initial Point
P
How can we find the direction? (Is this all
looking familiar for each application? You can
make a right triangle and use trig to get the
angle!)
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Q
Although it is possible to do this for any
initial and terminal points, since vectors are
equal as long as the direction and magnitude are
the same, it is easiest to find a vector with
initial point at the origin and terminal point
(x, y).
Terminal Point
A vector whose initial point is the origin is
called a position vector
direction is this angle
Initial Point
P
If we subtract the initial point from the
terminal point, we will have an equivalent vector
with initial point at the origin.
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To add vectors, we put the initial point of the
second vector on the terminal point of the first
vector. The resultant vector has an initial
point at the initial point of the first vector
and a terminal point at the terminal point of the
second vector (see below--better shown than put
in words).
To add vectors, we put the initial point of the
second vector on the terminal point of the first
vector. The resultant vector has an initial
point at the initial point of the first vector
and a terminal point at the terminal point of the
second vector (see below--better shown than put
in words).
To add vectors, we put the initial point of the
second vector on the terminal point of the first
vector. The resultant vector has an initial
point at the initial point of the first vector
and a terminal point at the terminal point of the
second vector (see below--better shown than put
in words).
Terminal point of w
Move w over keeping the magnitude and direction
the same.
Initial point of v
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The negative of a vector is just a vector going
the opposite way.
A number multiplied in front of a vector is
called a scalar. It means to take the vector and
add together that many times.
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Using the vectors shown, find the following
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Vectors Worksheet 1 Head-Minus Tail Rule   Prove
that the two vectors RS and PQ are
equivalent.   1.) R (-4, 7) S (-1, 5) P (0,
0) Q (3, -2)   2.) R (7, -3) S (4, -5) P
(0, 0) Q (-3, -2)   3.) R (2, 1) S (0,
-1) P (1, 4) Q (-1, 2)   4.) R (-2, -1) S
(2, 4) P (-3, -1) Q (1, 4)  
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This is the notation for a position vector. This
means the point (a, b) is the terminal point and
the initial point is the origin.
Vectors are denoted with bold letters
We use vectors that are only 1 unit long to build
position vectors. i is a vector 1 unit long in
the x direction and j is a vector 1 unit long in
the y direction.
(a, b)
(3, 2)
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If we want to add vectors that are in the form ai
bj, we can just add the i components and then
the j components.
When we want to know the magnitude of the vector
(remember this is the length) we denote it
Let's look at this geometrically
Can you see from this picture how to find the
length of v?
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Vectors Worksheet 2 Using the following
vectors P (-2, 2) Q (3, 4) R (-2, 5) S
(2, -8) Find PQ RS QR PS   2QS (v2)PR
3QR PS PS 3PQ
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Vector Worksheet 3 Performing Vector
Operations Let u lt-1, 3gt and v lt4, 7gt
Find the component form of the following
vectors. u v 3u 2u (-1)v   u v lt-1,
3gt lt4, 7gt lt-1 4, 3 7gt lt3, 10gt   3u
3lt-1, 3gt lt-3, 9gt   2u (-1)v 2lt-1, 3gt
(-1)lt4, 7gt lt-2, 6gt lt-4, -7gt lt-6, -1gt  
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Performing Vector Operations Now its your
turn! Let u lt-1, 3gt , v lt2, 4gt and w
lt2, -5gt Find u v u (-1)v u
w 3v 2u 3w 2u 4v 2u 3v u v
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Performing Vector Operations Now its your
turn! Let u lt-1, 3gt , v lt2, 4gt and w
lt2, -5gt Find u v lt1, 7gt u (-1)v
lt-3, -1gt u w lt-3, 8gt 3v lt6, 12gt 2u
3w lt4, -9gt 2u 4v lt-10, -10gt 2u
3v lt-4, -18gt u v lt-1, -7gt
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Unit Vectors and Direction Angles

• Any vector can be broken down into its
  components a horizontal component and a vertical
  component.
• In addition, any vector can be written as an
  expression in terms of a standard unit vector.
• Unit vectors help us separate vectors into
  componentsa scalar and a unit vector.

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Unit Vectors and Direction Angles

• The standard unit vectors are i and j.
• i lt1, 0gt
• j lt0, 1gt
• Using this style, we can now express vectors as a
  linear combination.

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Unit Vectors and Direction Angles

• Vector v lta, bgt
• lta, 0gt lt0, bgt
• alt1, 0gt blt0, 1gt
• We now have linear combination.
• v ai bj

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Unit Vectors and Direction Angles

• In vector v ai bj
• a and b are now scalars and express the
  horizontal and vertical components of vector v.
• We can use Trigonometry to calculate a direction
  angle for our vector.

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A unit vector is a vector with magnitude 1.
If we want to find the unit vector having the
same direction as a given vector, we find the
magnitude of the vector and divide the vector by
that value.
If we want to find the unit vector having the
same direction as w we need to divide w by 5.
Let's check this to see if it really is 1 unit
long.
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If we know the magnitude and direction of the
vector, let's see if we can express the vector in
ai bj form.
As usual we can use the trig we know to find the
length in the horizontal direction and in the
vertical direction.
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Acknowledgement I wish to thank Shawna Haider
from Salt Lake Community College, Utah USA for
her hard work in creating this PowerPoint. www.sl
cc.edu Shawna has kindly given permission for
this resource to be downloaded from
www.mathxtc.com and for it to be modified to suit
the Western Australian Mathematics Curriculum.
Stephen Corcoran Head of Mathematics St
Stephens School Carramar www.ststephens.wa.edu.
au