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Learning Outcomes At the end of the lesson you should
be able to:1. Identify a vector in 3- space from
2-space and 1-space.2. Find the distance between points.3. Derive the equation of sphere.4. Define vector.5. Perform the arithmetic of vectors.6. Perform Dot and Cross product
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Rectangular Coordinate Systems
One way to identify points in a plane is to use a Cartesian coordinate system.
3- space: x , y, and z axes 2- space: x and y axes produces a Plane 1-space : x axes, a linez
y
xo
y
xo
xo
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OCTANTS
z
y
x ab
cP(a, b, c)
Z-axis : (0, 0, c)Y-axis : (0,b, 0 )X-axis : (a, 0, 0)
The coordinate axes define 3 coordinate plates: xy-plane, yz-plane and xz-plane.
The location of point P in space can be specified by listing 3 real numbers:
Mid-point The coordinate of the center between points A(a,b) and B(c,d) is given by
The coordinate of the point divides points A(a,b) and B(c,d) in a ratio m:n is given by
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2,2dbcaM
nmdmbn
nmcman ,
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Distance (d) in 3-space
22 ba
222 cbad
),,( 1111 zyxP
d c
ba
y
),,( 2222 zyxP
212
212
212 )()()( zzyyxxd
czzbyyaxx 121212 ;;
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Theorem: An equation of the form
where E, F, G and H are constant, represents a sphere , a point , or has no graph.
0222 HGzFyExzyx
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Example 1Find the center and radius of the sphere that has (1, -2, 4) and ( 3, 4, -12) as endpoints of a diameter.Answers: C (h,k,l) =( 2, 1, -4) d =17.3 ; r = 8.6
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Example 2Show that A(4, 5,2),B(1,7,3) and C(2,4,5) are vertices of an equilateral triangle.
AB=AC=BC= 14Answer:
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Simple Recall1. Find the distance between the
points A ( 2, 1, 3) and B (-1,4,1). 2. Find the equation of the sphere
with center at the origin and passing through a point (-1,-1, 2).
3. Find the center and radius of the sphere given
11246222 zyxzyx
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VECTORS Definition Vector is a quantity that has
both magnitude and direction usually represented by an arrow.
It has an initial point and a terminal point.Vector quantity: velocity,
displacement, force
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How to Draw a Vector? Example 1: v= <1, 3, 2> in 3-dimensional
Example 2: v= <4, -1> in 2-dimensional
Example 3 : a= 2i - 3j + 4k where i, j, k are unit vectors
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Equivalent Vectors Two vectors are equivalent if their corresponding components are equal.
Example: A= <2,-1,3> and B= < 2, -1, 3>
Vector A is equivalent to vector B !
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Arithmetic Operations on Vectors Theorem. If
spaceinwvwvwvwvwwwwandvvvv
spaceforwvwvwvwwwandvvv
3,,,,,,
;2,,,
332211
321321
2211
2121
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If v is a nonzero vector and k is a nonzero real number (scalar) then the scalar multiple kv is defined to be a vector whose length is k times the length of v and whose direction is the same as that of v if k > 0 and opposite that of v if k< 0.
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spaceinwvwvwvwvspaceinwvwvwv
3,,2,332211
2211
If k is any scalar, then 32121 ,,, kvkvkvkvorkvkvkv
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Vectors with Initial Point Not the Origin Theorem.
space3in,,,ALSOspace2in,),(pointterminaland),(
pointinitialwithspace2invectoraisIf
12121221
121221
222111
21
zzyyxxPPyyxxPP
yxPyxPPP
lengththeis|| 21PP
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Norm of a Vector The norm or the magnitude (length) is denoted as
32123
22
21
2122
21
,,;space3in,;space2in
vvvvvvvv
vvvvvv
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Unit Vectors In 2-space: i =<1, 0 > ; j =<0 ,1> In 3-space: i=<1,0,0>; j=<0,1,0> ; k=<0, 0, 1>
i , j, k is known as the standard basis vector
Example: A= <-1,5,-6> = -i + 5j- 6z
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Normalizing a Vector vkvv
u 1
is a unit vector with the same direction as v.Example 4.Find the unit vector that has the same direction as v=2i-j+4kAnswer: 4,1,2
211u
Dot Product
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332211 vuvuvuvu
What is the dot product of u and v?
2,4,3say,,3,2,1say,,
321
321
vvvvvuuuuu
Note: the dot product of 2 vectors produces a scalar value
Cross Product
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21
21
31
31
32
32
321
321
321321 ,,&,,
vvuu
kvvuu
jvvuu
ivu
vvvuuukji
vu
vvvvuuuu
Note: The cross product of 2 vectors produces a vector!Cross Product is not commutative!
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