The 3D coordinate system

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The 3D coordinate system
Distance formula The distance P1P2 between the
points P1(x1, y1, z1) and P2(x2 , y2 , z2) is
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The 3D coordinate system
Mid-point of a line segment Let L be a line
segment with end points P1(x1, y1, z1) and P2(x2
, y2 , z2), then the coordinates of its midpoint
are
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Equation of a Sphere An equation of a sphere with
center at the origin and radius r is
An equation of a sphere with center at (a, b, c)
and radius r is
(x a)2 (y b)2 (z c)2 r2
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Vectors
Definition A vector is a physical quantity that
has both magnitude and direction.
Remark Two vectors u and v are considered to be
equal if they have the same direction and
magnitude. In particular, the position of a
vector is unimportant.
The zero vector 0 is the special vector that has
0 magnitude and no direction.
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Notations
Through out this course, vectors will be denoted
by bold face Italic letters such as u, v, and w.
Scalars will be denoted by light face Italic
letters such as a, b, and c.
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Addition of vectors If u and v are vectors
positioned in such a way that the initial point
of v coincides with the terminal point of u, then
the sum u v is defined to be the vector from
the initial point of u to the terminal point of v.
u v
v
u
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• Scalar Multiplication
• If c is a scalar and v is a vector, then the
  scalar multiple cv is the vector whose length is
  c times the length of v and whose direction is
• the same as v if c gt 0
• opposite to v if c lt 0

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Components of a vector If the vector v starts at
(x, y, z) and terminates at (x
a1, y a2, z a3), then the component form of
v is ?a1, a2, a3?
Unit vectors are vectors whose lengths are 1
unit. Three special unit vectors are
i ?1, 0, 0? , j ?0, 1, 0? , k ?0, 0, 1?
Hence any vector v ?a1, a2, a3? can be
written as
v a1 i a2 j a3 k
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Addition of vectors in component form If u ?a1,
a2, a3? and v ?b1, b2, b3?, then
u v ?a1 b1, a2 b2, a3 b3?
Scalar Multiplication If c is a scalar and u
?a1, a2, a3? is a vector, then
cu ?ca1, ca2, ca3?
Magnitude of a vector If u ?a1, a2, a3? is a
vector, then its magnitude is
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Properties of Vectors

• If u, v, w, are vectors in ?3, and c and d are
  scalars, then
• u v v u
• u (v w) (u v) w
• u 0 u
• u (- u) 0
• c(u v) cu cv
• (c d) u cu d u
• (c d) u c(d u)
• 1 u u

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Direction cosines of a vector Given a vector v
?a, b, c?, then its direction is determined by 3
angles, namely angle ? the angle between v and
the x direction angle ? the angle between v and
the y direction angle ? the angle between v and
the z direction (all these angles have measures
between 0o and 180o) and we have the following
relations,
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The Dot Product
Geometrical definition If u ?a1, a2, a3? and v
?b1, b2, b3 ? are vectors, then their dot
product is defined to be
u v uvcos ? where
? is the angle between u and v.
Algebraic definition If u ?a1, a2, a3? and v
?b1, b2, b3 ? are vectors, then their dot product
is defined to be u v
a1b1 a2 b2 a3 b3
Remark The algebraic definition allows the dot
product to be defined on vector spaces of higher
dimensions.
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The angle ? between two vectors u and v can be
computed by the formula
Corollary Two vectors u and v are orthogonal if
and only if u v 0 Two
vectors u and v are in the same direction if and
only if u v uv
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Properties of the Dot Product

• If u, v, w are vectors in ?3, and c is a scalar,
  then
• u u u2
• u v v u
• u (v w) (u v) (u w)
• (cu) v c(u v) u (cv)
• 0 u 0

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Projections
Let u and v be two vectors in the same space,
then the vector projection of v onto u is

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On Jan 24, 2006, a 1994 Cessna Citation V
twin-engine jet coming from Hailey, Idaho, near
the resort community of Sun Valley, around 640
a.m. skidded off the end of Runway 24 at Palomar
airport killing 4 on board. Strong Santa Ana wind
was blowing from the east that morning and the
pilot chose the wrong runway of land with the
wind. The plane touched down beyond the midpoint
of the runway and did not have enough distance to
stop because of the excessive ground speed.
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Cross Product
DefinitionGiven two vectors u ?a1, a2, a3? and
v ?b1, b2, b3? in 3D space, the cross product
of u and v is the vector u v ?
a2 b3 - a3 b2 , a3 b1 - a1 b3 , a1 b2 - a2 b1 ?
If this is hard to remember, we can use the
determinant form
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Geometric definition of the cross product If ?
(between 0 and ?) is the angle between u and v,
then u v
uvsin? i.e. the magnitude is the area of the
parallelogram form by the vectors u and v.
v
?
u
The direction of uv is perpendicular to both u
and v and is determined by the right-hand rule.
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Definition Two vector are said to be parallel if
the angle between them has measure equal to 0 or
?.
Theorem Two vectors u and v in 3D space are
parallel if and only if
u v 0 v u
An easier way to check that two vectors are
parallel is to check whether one is a scalar
multiple of another.
Remark The cross product is only defined for
vectors in 3D space, It cannot be extended to
higher dimensional spaces.
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Properties of the Cross product

• Theorem
• If u, v, w are vectors in 3D space and c is a
  scalar, then
• u v - v u
• (c u) v c(u v) u (c v)
• u (v w) u v u w
• (u v) w u w v w
• u (v w) (u v) w
• u (v w) (u w)v (u v)w

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The scalar triple product
Given three vectors u, v, w in 3D space, the
product u
(v w) ( or (u v) w ) is called
the triple product (also called the box product)
of the three vectors. The
geometrical meaning is that u (v
w) the volume of the parallelopiped
form by the three
vectors u, v, and w.
parallelopiped
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Corollary Three vectors u, v, w in 3D space are
coplanar (i.e. lying on the same plane if they
start from the same point) if and only if
u (v w) 0
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Equations of Lines and Planes
A line is a 1 dimensional object in space, hence
it should have only 1 variable, but we need two
points to specify its position and also
direction. We will start with a line segment
with two given end points.
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Vector equation of a line segment from a point r0
to another point r1 is r(t) (1
t)r0 t r1 where t ranges from 0 to 1.
r0
r1
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Vector equation of a line passing through two
points r0 and r1 is r(t) (1
t)r0 t r1 where t ranges from -8 to 8 .
r0
r1
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Vector Equation of a Line Suppose that v is a
direction vector of the line, r0 is the position
vector of a given point on the line, then the
position vector r of a generic point on the line
is
r r0 t v where t is the
parameter that ranges from -8 to 8
t v
r0
r
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Parametric Equations of a Line Suppose that the
direction vector in the previous equation is v
?a, b, c?, and r ?xo, yo, zo?, then the
position of a generic point on the line can be
specified by the following set of equations x
xo at, y yo bt, z zo
ct where t is parameter (variable).
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Symmetric Equations of a Line If we eliminate the
parameter t from the previous set of parametric
equations, we then get another set of equations
for the same line,
And we call a, b, c, the direction numbers of
this line because they are the components of a
direction vector for this line.
Remark This type of equations requires that all
direction numbers to be non-zero.
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Definition Two lines in the 3D space is said to
be a pair of skew lines if they are not parallel
and they do not intersect (hence they cannot lie
on the same plane).
Remark Two lines are parallel if and only of they
have direction vectors that are scalar multiples
of each other.
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Planes
A plane is a 2 dimensional object in space and
hence its equation should have 2 parameters, and
we need 3 (non-colinear) points to completely
specify a plane.
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Normal vectors of a plane
A plane W does not have a direction, but it is
perpendicular to two (opposite) directions in the
sense that there are two unit vectors n and n
that are perpendicular to any vector v lying on
the plane W. These two vectors are called
(unit) normal vectors to the plane W.
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• Useful criteria
• Two planes are parallel if and only if they have
  parallel normal vectors.
• Two planes are perpendicular if and only if they
  have perpendicular normal vectors.

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Vector Equation of a plane If n ?a, b, c? is
a normal vector (not necessary of unit length) of
a plane and P(x0, y0, z0) is a point on the
plane, then an equation for the plane is
n ?x x0, y y0, z
z0 ? 0 or a(x x0 ) b(y y0)
c(z z0) 0
Linear equation of a plane If we rearrange the
terms in the above equation, it will change to
the form ax by cz d
0 where d -(ax0 by0 cz0)
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More on Linear Equations

• Suppose that we are given a linear equation
• ax by cz d 0
• for a plane, then
• ?a, b, c? will be a normal vector to the plane
• if a ? 0, then -d /a is the x-intercept of the
  plane
• if b ? 0, then -d /b is the y-intercept of the
  plane
• if c ? 0, then -d /c is the z-intercept of the
  plane

Conversely, if we know that the intercepts of the
plane are a, ß, and ? respectively, then an
equation for the plane is
provided that a, ß, and ? are all non-zero.
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Lines from Planes
It is not hard to see that any two non-parallel
planes will intersect and their intersection must
be a line. Hence a line can be specified by the
equations of two (non-parallel) planes.
a1x b1y c1z d1 0
a2x b2y c2z d2 0
A direction vector for this line can be computed
by v ? a1, b1, c1 ? ? a2,
b2, c2 ?
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Distance D from a point P1(x1, y1, z1) to the
plane ax by cz d 0
is given by the formula
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Some Typical Problems

• Given a line L and a point P not on the line,
  find the equation of the line L2 that passes
  through P and runs parallel to L.
• Given a line L and a point P not on L, find the
  shortest distance from P to L.
• Find the minimum distance between two given
  parallel lines.

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Some Typical Problems

• Given two intersecting planes and a point P not
  on any of these two planes, find an equation of
  the line that passes through P and runs parallel
  to the intersection of the two planes.
• Given a plane W and a point P not on W, find the
  shortest distance from P to W.
• Given a plane W and a point P not on W, find an
  equation for the plane W2 that passes through P
  and is parallel to W.

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Some Typical Problems

• Find the shortest distance between two given
  parallel planes.
• Given a line L and a point P not on L, find an
  equation for the plane containing P and L.
• Find the shortest distance between two given skew
  lines.