Initial point

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terminal point
B(x2 , y2)
v
Initial point
A(x1 , y1)
VAB, vector from the point A to the point B, is
a directed line segment between A and B.
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MAGNITUDE OF VECTOR
B(x2 , y2)
v
A(x1 , y1)
V (x2-x1)2 (y2-y1)2
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EQUIVALENT VECTORS
SAME MAGNITUDE SAME DIRECTION
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UNIT VECTOR
U
U 1
MAGNITUDE OF VECTOR IS 1
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EXAMPLES
1- Vector v has the initial point A(3, 4) and
the terminal point B( -2, 5).
Magnitude of vector v is (-2-3)2 (5-4)2
(-5)2 12 251 26
V 26
V is not a unit vector
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2- Vector i has the initial point (0, 0) and the
terminal point(1, 0). Magnitude of the vector i
is (1-0)2 (0-0)2
12 02 1 0 1
i
i is a unit vector
3- Vector j has the initial point (0, 0) and the
terminal point (0 ,1). Magnitude of the vector j
is (0-0)2 (1-0)2
12 02 1 0 1
j
j is a unit vector
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SCALAR MULTIPLICATION OF VECTOR
V
4k
k
0.5V
W
-W
2W
-2W
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VECTOR ADDITION
WV
V
W
TRIANGLE METHOD
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VECTOR ADDITION
V
WV
W
PARALLELOGRAM METHOD
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VECTOR ADDITION
V
WV
V-W
W
-W
W-V
-V
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VECTORS IN A COORDINATE PLANE
y
C(x2-x1 , y2-y1)
B(x2 , y2)
V
WAB
x
O
V x2-x1 , y2-y1
A(x1 , y1)
Vectors W and V are equivalent vectors. Since V
starts from the origin we use a special notation
for V
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VECTORS IN A COORDINATE PLANE
y
B(1 , 4)
4
C(-4 , 3)
W AB
3
V
A(5 , 1)
1
x
O
5
-4
1
V -4 , 3
Vectors W and V are equivalent vectors.
-4 is the x-component ,and 3 is the y-component
of the vector V
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VECTORS IN A COORDINATE PLANE
y
A(1 , 4)
4
W AB
B(-3 , 1)
1
-4
x
- 3
1
V
Vectors W and V are equivalent vectors.
C(-4 , -3)
- 3
V -4 , -3
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VECTORS IN A COORDINATE PLANE
y
j 0 , 1
1
j
i
x
1
i 1 , 0
Vectors i and j are special unit vectors.
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VECTORS IN A COORDINATE PLANE
Find a vector that has the initial point (3 , -1)
and is equivalent to V -2 , 3 .
y
If ( x, y) is the terminal point of W, then x-3
-2 ? x 1 and y-(-1) 3 ? y 2
P(-2 , 3)
3
B( 1 ,2)
V
2
W
3
x
1
- 2
W AB
- 1
A(3 , -1)
Vectors W and V are equivalent vectors.
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BASIC VECTOR OPERATIONS
V a , b and W c , d are two
vectors and k is a real number.
1- V a2 b2 2- vw
a , b c , d ac , bd 3-
kV k a , b ka , kb 4-
kV k V
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BASIC VECTOR OPERATIONS
V -2 , 3 , W 4 , -1
v (-2)2 32 4 9 13
V W -2 , 3 4 , -1
-24, 3-1 2 , 2
5V 5 -2 , 3 -10 , 15
-3W -3 4 , -1 -12 , 3
5V -3W -10 , 15 -12 , 3
-10-12 , 153 -22 , 18
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ANY VECTOR CAN BE WRITTEN IN TERMS OF THE UNIT
VECTORS i AND j
If V a , b is any vector, then by
using basic vector operations we get V a ,
b a , 0 0 , b
a 1 , 0 b 0 , 1
ai bj
V a , b ai bj
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VECTORS WRITTEN IN TERMS OF THE UNIT VECTORS i
AND j
y
P(4 , 3)
3j
4i3j
j
i
4i
x
V 4i3j 4 , 3
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VECTORS WRITTEN IN TERMS OF THE UNIT VECTORS i
AND j
y
j
i
4i
x
4i-3j
-3j
P(4 , -3)
V 4i-3j 4 , -3
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DIRECTION ANGLE OF VECTORS
x
cosa
y
V
P(x , y)
y
sina
V
V
ß
a direction angle of V
W
x
Q(a , b)
V x , y V cosa , sina
V
cosa ,
V
sina
W a , b W cosß , sinß
cosß ,
sinß
W
W
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DIRECTION ANGLE OF VECTORS
If V -2i 3j , then find the direction angle
of V.
V -2i3j -2 , 3
y
3
3
tana -
x
-2
2
3
a tan-1- in the second quadrant
2
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DIRECTION ANGLE OF VECTORS
1
3
If V , - , then find
the direction angle of V.
2
2
3
y
2
tana - -
3
x
1
2
p
a tan-1- -
3
3
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DIRECTION ANGLE OF VECTORS
7p
If V 6 and the direction angle of V i s
, then find the x and y-components of V.
6
cosa ,
V x , y
sina
V
V
3
1
7p
7p
V
6cos ,
6sin
-6 ,
-6
6
6
2
2
V
-3 3 , -3
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UNIT VECTORS ON THE SAME DIRECTION WITH A GIVEN
VECTOR
If V x , y , then
x
y
U ,
cos? ,
sin?
V
V
is the unit vector on the same direction with V.
? is the direction angle of V
V x , y
U
V
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UNIT VECTORS ON THE SAME DIRECTION WITH A GIVEN
VECTOR
If V -3 , 4 , then find the unit vector
on the same direction.
v (-3)2 42 9 16 25
5
-3
4
x
y
U ,
,
5
5
V
V
Now find the vector W on the same direction with
magnitude 6.
-3
4
-18
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W 6U 6
See the illustrations on the
next slide
,
,
5
5
5
5
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y
W 6U
W 6
W
4
P(-3 , 4)
V
V 5U
V 5
U
x
O
-3
X
X -3U
X3
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EXAMPLE
If V -2i 4j , then find the vector on the
opposite direction with magnitude 6. First,
find the unit vector on the direction of V.
U
, Now, multiply the unit
vector with -6. That will give you the answer.
W
-6U The vector on the opposite direction with
magnitude 6.
x
y
-2
4
20
20
V
V
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DOT PRODUCT OF VECTORS
V a , b and W c , d
VW ac bd
VV a2 b2 V2
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DOT PRODUCT OF VECTORS
1- VW WV dot product is commutative

2- U(V W) UV UW distributive

3- a (VW) (aV )WV(aW) ,a is a scalar

4- VV V2
5- 0W 0 zero vector

6- ii jj 1
7- ij ji 0
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DOT PRODUCT OF VECTORS
UV2 (UV).(UV) U.UU.VV.UV.V
UV2 U2 2U.V V2
SIMILARLY
U-V2 (U-V).(U-V) U.U-U.V-V.UV.V
U-V2 U2 - 2U.V V2
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ANGLE BETWEEN TWO VECTORS
V
W
?
? angle between V and W
VW
VW VWcos ?
cos ? __________
VW
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EXAMPLE
V -2 , 3 , W 4 , - 1
VW ac bd -2.4 3.(- 1) - 8 - 3 -11
VV a2 b2 (-2)2 32 13 V2
VW
-11
cos ? __________ _______
VW
13 17
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EXAMPLE
V 3 , -6 , W -1 , 2
VW 3. (-1) - 6.2 - 3-12 -15
V 45 , W 5
VW
-15
-15
-15
cos ? __________ _______ ______ _____
-1
VW
45 5
225
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cos ? -1, then ? cos-1(-1) p
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EXAMPLE
V
?
W
If VW 0 ,then V and W are perpendicular
? is 90? , cos ? 0
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EXAMPLE
If V 4i-3j, then find a vector that is
perpendicular to V
If W xi yj, then
VW 4x 3y 0
Any choice of x and y that satisfies the equation
above is an answer
Since 3 and 4 satisfy the equation
W 3, 4 is one of the vectors
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V and W are parallel
W
Same direction
V
W
? is 0?, cos ? 1
V
Opposite direction
? is 180?, cos ? -1
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EXAMPLE
V 3 , -6 , W -1 , 2
VW
-15
-15
-15
cos ? __________ _______ ______ _____
-1
VW
45 5
225
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cos ? -1, then ? cos-1(-1) p
V and W are parallel with opposite direction
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EXAMPLE
V 3 , -6 , W 1 , -2
VW
15
15
15
cos ? __________ _______ ______ ___ 1
VW
45 5
225
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cos ? 1, then ? cos-1(1) 0
V and W are parallel with same direction
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EXAMPLE
V 3 , -6 , W 4 , 2
VW
0
cos ? __________ _______ 0
VW
45 20
cos ? 0 , then ? cos-1(0) 90?
V and W are perpendicular ( orthogonal ) vectors
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EXAMPLE
1
If U V 7 , U is a unit vector and cos ?
____ where ? is the angle between U and V,
then find the magnitude of V.
2
UV2 U2 2U.V V2 7
1 2U.V V2 7 , V2 2U.V - 6 0
cos ? __________ ___ , 2U.V V
UV
1
2
UV
V2 2V - 6 0 (V-2)(V3) 0
V 2 or V -3, V 0 so V 2
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SCALAR PROJECTION
W
V
?
VW
projWV Vcos ? _______

W
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EXAMPLE
If V 2i 2j , and W -4i-2j , then find
projWV
projWV _______ _____ ______ ____


-12
-12
-6
VW
W
20
2 5
5
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EXAMPLE
If V kW for any nonzero number k , then
VW
(kW)W
k(WW)
kW2
projWV _______ _________ _________
________

W
W
W
W
projWV kW
How about projVW ?.