Vectors
Data Engineering
Linear Algebra
1 / 24
Vectors
Definition. Properties. Operations
Vector = quantity that has both magnitude (or length), and
direction.
Data Engineering
Linear Algebra
2 / 24
Vectors
Definition. Properties. Operations
Vector = quantity that has both magnitude (or length), and
direction.
Two vectors are equal if they have the same magnitude and
direction.
Data Engineering
Linear Algebra
2 / 24
Vectors
Definition. Properties. Operations
Vector = quantity that has both magnitude (or length), and
direction.
Two vectors are equal if they have the same magnitude and
direction.
Quantities that have only magnitude but no direction are called
scalars.
Zero vector, 0, is the only exception - it has magnitude
(length) 0, but no direction.
Data Engineering
Linear Algebra
2 / 24
Vectors
Definition. Properties. Operations
Multiplication by a scalar
Given a scalar c and a vector v, the vector cv is |c| times the length
of v. If c > 0, then cv has the same direction as v. If c < 0, then cv
points in the opposite direction to v.
Two vectors are parallel if they have the same direction, that is if
one vector is a scalar multiple of the other.
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Linear Algebra
3 / 24
Vectors
Definition. Properties. Operations
Vector Addition
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Linear Algebra
4 / 24
Vectors
Definition. Properties. Operations
Vector Addition
Triangle law
Pallalelogram law
Data Engineering
Linear Algebra
4 / 24
Vectors
Definition. Properties. Operations
Vector coordinates
The vector v with its tail at (0, 0) and its head at the point (vx , vy ) is
called the position vector, and is denoted by [vx , vy ]. The real
numbers vx and vy are called the coordiantes of v.
Data Engineering
Linear Algebra
5 / 24
Vectors
Definition. Properties. Operations
Vector coordinates
The vector v with its tail at (0, 0) and its head at the point (vx , vy ) is
called the position vector, and is denoted by [vx , vy ]. The real
numbers vx and vy are called the coordiantes of v.
−−→
Vector P Q with its tail at the point P (x1 , y1 ) and head at Q(x2 , y2 )
has coordiantes
−−→
P Q = [x2 − x1 , y2 − y1 ]
Data Engineering
Linear Algebra
5 / 24
Vectors
Definition. Properties. Operations
Vector coordinates
The vector v with its tail at (0, 0) and its head at the point (vx , vy ) is
called the position vector, and is denoted by [vx , vy ]. The real
numbers vx and vy are called the coordiantes of v.
−−→
Vector P Q with its tail at the point P (x1 , y1 ) and head at Q(x2 , y2 )
has coordiantes
−−→
P Q = [x2 − x1 , y2 − y1 ]
Operations on vectors
Let u = [ux , uy ] and v = [vx , vy ]
u = v if and only if ux = vx and uy = vy
u + v = [ux + vx , uy + vy ]
cv = [cvx , cvy ]
Data Engineering
Linear Algebra
5 / 24
Vectors
Definition. Properties. Operations
Magnitude/Length of a vector
−−→
−−→
Length of the vector P Q = [x2 − x1 , y2 − y1 ], denoted by |P Q|, is
the distance between the points P and Q
p
−−→
|P Q| = (x2 − x1 )2 + (y2 − y1 )2
Data Engineering
Linear Algebra
6 / 24
Vectors
Definition. Properties. Operations
Magnitude/Length of a vector
−−→
−−→
Length of the vector P Q = [x2 − x1 , y2 − y1 ], denoted by |P Q|, is
the distance between the points P and Q
p
−−→
|P Q| = (x2 − x1 )2 + (y2 − y1 )2
A unit vector is any vector with length 1. A unit vector in the
direction of v is a vector u given by
u=
Data Engineering
v
|v|
Linear Algebra
6 / 24
Vectors
Definition. Properties. Operations
Magnitude/Length of a vector
−−→
−−→
Length of the vector P Q = [x2 − x1 , y2 − y1 ], denoted by |P Q|, is
the distance between the points P and Q
p
−−→
|P Q| = (x2 − x1 )2 + (y2 − y1 )2
A unit vector is any vector with length 1. A unit vector in the
direction of v is a vector u given by
u=
v
|v|
Versors of the coordinate system OXY , denoted by i = [1, 0] and
j = [0, 1], are unit vectors in the direction of the coordinate axes.
Data Engineering
Linear Algebra
6 / 24
Vectors
3D Cartesian system
3D coordinate system
A three-dimensional coordinate system, OXYZ = three axes
perpendicular to each other with a common unit, intersecting at the
origin, which has coordinates (0, 0, 0).
right-handed coordinate system
Data Engineering
Linear Algebra
7 / 24
Vectors
3D Cartesian system
Three coodinate planes, OXY, OXZ, OYZ, divide the xyz−space into
eight regions called octants.
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Linear Algebra
8 / 24
Vectors
3D Cartesian system
A point P in space = ordered triple of numbers, so called
coordinates of the point P (xP , yP , zP )
Data Engineering
Linear Algebra
9 / 24
Vectors
3D Cartesian system
A point P in space = ordered triple of numbers, so called
coordinates of the point P (xP , yP , zP )
Distance between the points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 )
p
|AB| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2
Data Engineering
Linear Algebra
9 / 24
Vectors
Vectors in 3D
Vectors in 3D
position vector: v = [vx , vy , vz ] has its tail at (0, 0, 0) and head
at (vx , vy , vz ).
Data Engineering
Linear Algebra
10 / 24
Vectors
Vectors in 3D
Vectors in 3D
position vector: v = [vx , vy , vz ] has its tail at (0, 0, 0) and head
at (vx , vy , vz ).
−−→
vector coordinates: P Q = [x2 − x1 , y2 − y1 , z2 − z1 ]
Data Engineering
Linear Algebra
10 / 24
Vectors
Vectors in 3D
Vectors in 3D
position vector: v = [vx , vy , vz ] has its tail at (0, 0, 0) and head
at (vx , vy , vz ).
−−→
vector coordinates: P Q = [x2 − x1 , y2 − y1 , z2 − z1 ]
p
−−→
length: |P Q| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2
Data Engineering
Linear Algebra
10 / 24
Vectors
Vectors in 3D
Vectors in 3D
position vector: v = [vx , vy , vz ] has its tail at (0, 0, 0) and head
at (vx , vy , vz ).
−−→
vector coordinates: P Q = [x2 − x1 , y2 − y1 , z2 − z1 ]
p
−−→
length: |P Q| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2
equality: u = v if and only if ux = vx , uy = vy oraz uz = vz
Data Engineering
Linear Algebra
10 / 24
Vectors
Vectors in 3D
Vectors in 3D
position vector: v = [vx , vy , vz ] has its tail at (0, 0, 0) and head
at (vx , vy , vz ).
−−→
vector coordinates: P Q = [x2 − x1 , y2 − y1 , z2 − z1 ]
p
−−→
length: |P Q| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2
equality: u = v if and only if ux = vx , uy = vy oraz uz = vz
addition: u + v = [ux + vx , uy + vy , uz + vz ]
multiplication by a scalar: cv = [cvx , cvy , cvz ]
Data Engineering
Linear Algebra
10 / 24
Vectors
Vectors in 3D
Vectors in 3D
position vector: v = [vx , vy , vz ] has its tail at (0, 0, 0) and head
at (vx , vy , vz ).
−−→
vector coordinates: P Q = [x2 − x1 , y2 − y1 , z2 − z1 ]
p
−−→
length: |P Q| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2
equality: u = v if and only if ux = vx , uy = vy oraz uz = vz
addition: u + v = [ux + vx , uy + vy , uz + vz ]
multiplication by a scalar: cv = [cvx , cvy , cvz ]
versors: i = [1, 0, 0] , j = [0, 1, 0] and k = [0, 0, 1]
[vx , vy , vz ] = vx i + vy j + vz k
Data Engineering
Linear Algebra
10 / 24
Vectors
Dot product
A dot (scalar) product of nonzero vectors u and v
Data Engineering
Linear Algebra
11 / 24
Vectors
Dot product
A dot (scalar) product of nonzero vectors u and v
u ◦ v = |u| · |v| · cos ∠(u, v)
Data Engineering
Linear Algebra
11 / 24
Vectors
Dot product
A dot (scalar) product of nonzero vectors u and v
u ◦ v = |u| · |v| · cos ∠(u, v)
u ◦ v = ux vx + uy vy + uz vz
Data Engineering
Linear Algebra
11 / 24
Vectors
Dot product
A dot (scalar) product of nonzero vectors u and v
u ◦ v = |u| · |v| · cos ∠(u, v)
u ◦ v = ux vx + uy vy + uz vz
Consequence
u⊥v
Data Engineering
⇐⇒
u◦v =0
Linear Algebra
11 / 24
Vectors
Dot product
A dot (scalar) product of nonzero vectors u and v
u ◦ v = |u| · |v| · cos ∠(u, v)
u ◦ v = ux vx + uy vy + uz vz
Consequence
u⊥v
⇐⇒
u◦v =0
Properties of the dot product
u ◦ u = |u|2
Data Engineering
Linear Algebra
11 / 24
Vectors
Dot product
A dot (scalar) product of nonzero vectors u and v
u ◦ v = |u| · |v| · cos ∠(u, v)
u ◦ v = ux vx + uy vy + uz vz
Consequence
u⊥v
⇐⇒
u◦v =0
Properties of the dot product
u ◦ u = |u|2
u◦v =v◦u
Data Engineering
Linear Algebra
11 / 24
Vectors
Dot product
A dot (scalar) product of nonzero vectors u and v
u ◦ v = |u| · |v| · cos ∠(u, v)
u ◦ v = ux vx + uy vy + uz vz
Consequence
u⊥v
⇐⇒
u◦v =0
Properties of the dot product
u ◦ u = |u|2
u◦v =v◦u
→
−
u ◦ u > 0 for u 6= 0
Data Engineering
Linear Algebra
11 / 24
Vectors
Dot product
A dot (scalar) product of nonzero vectors u and v
u ◦ v = |u| · |v| · cos ∠(u, v)
u ◦ v = ux vx + uy vy + uz vz
Consequence
u⊥v
⇐⇒
u◦v =0
Properties of the dot product
u ◦ u = |u|2
u◦v =v◦u
(αu) ◦ v = u ◦ (αv) = α(u ◦ v)
→
−
u ◦ u > 0 for u 6= 0
Data Engineering
Linear Algebra
11 / 24
Vectors
Dot product
A dot (scalar) product of nonzero vectors u and v
u ◦ v = |u| · |v| · cos ∠(u, v)
u ◦ v = ux vx + uy vy + uz vz
Consequence
u⊥v
⇐⇒
u◦v =0
Properties of the dot product
u ◦ u = |u|2
u◦v =v◦u
→
−
u ◦ u > 0 for u 6= 0
Data Engineering
(αu) ◦ v = u ◦ (αv) = α(u ◦ v)
u ◦ (v + w) = u ◦ v + u ◦ w
Linear Algebra
11 / 24
Vectors
Projection
Orthogonal projections
An orthogonal (perpendicular) projection of the vector u
onto vector v is the vector
v
u ◦ v
uv = |u| · cos α ·
=
·v
|v|
v◦v
Data Engineering
Linear Algebra
12 / 24
Vectors
Cross product
Cross product
Data Engineering
Linear Algebra
13 / 24
Vectors
Cross product
Cross product
A cross (vector) product of two nonzero vectors u and v is the
vector u × v perpendicular to both u and v with length
|u × v| = |u| · |v| · sin ∠(u, v)
given by the formula
−
→
− →
− →
i
j k
u × v = ux uy uz = [uy vz − uz vy , uz vx − ux vz , ux vy − uy vx ]
vx vy vz
Data Engineering
Linear Algebra
13 / 24
Vectors
Cross product
Consequence
ukv
Data Engineering
⇐⇒
→
−
u×v = 0
Linear Algebra
14 / 24
Vectors
Cross product
Consequence
ukv
⇐⇒
→
−
u×v = 0
Properties
u × v = −(v × u)
Data Engineering
Linear Algebra
14 / 24
Vectors
Cross product
Consequence
ukv
⇐⇒
→
−
u×v = 0
Properties
u × v = −(v × u)
u × (v + w) = u × v + u × w
(αu) × v = u × (αv) = α(u × v)
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Linear Algebra
14 / 24
Vectors
Mixed product
A mixed product of three vectors
Data Engineering
Linear Algebra
15 / 24
Vectors
Mixed product
A mixed product of three vectors
ux uy uz
u ◦ (v × w) = vx vy vz
wx wy wz
Properties
Data Engineering
Linear Algebra
15 / 24
Vectors
Mixed product
A mixed product of three vectors
ux uy uz
u ◦ (v × w) = vx vy vz
wx wy wz
Properties
u ◦ (v × w) = v ◦ (w × u) = w ◦ (u × v)
Data Engineering
Linear Algebra
15 / 24
Vectors
Mixed product
A mixed product of three vectors
ux uy uz
u ◦ (v × w) = vx vy vz
wx wy wz
Properties
u ◦ (v × w) = v ◦ (w × u) = w ◦ (u × v)
u ◦ (v × w) = −u ◦ (w × v)
Data Engineering
Linear Algebra
15 / 24
Vectors
Mixed product
A mixed product of three vectors
ux uy uz
u ◦ (v × w) = vx vy vz
wx wy wz
Properties
u ◦ (v × w) = v ◦ (w × u) = w ◦ (u × v)
u ◦ (v × w) = −u ◦ (w × v)
Vectors u, v, w are coplanar ⇐⇒ u ◦ (v × w) = 0
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Linear Algebra
15 / 24
Vectors
Applications
Geometrical applications of vector products
1
Area of the parallelogram with sides u and v
AP = |u × v|
Data Engineering
Linear Algebra
16 / 24
Vectors
Applications
Geometrical applications of vector products
1
Area of the parallelogram with sides u and v
AP = |u × v|
2
Area of the triangle with sides u and v
1
A4 = |u × v|
2
Data Engineering
Linear Algebra
16 / 24
Vectors
Applications
Geometrical applications of vector products
1
Area of the parallelogram with sides u and v
AP = |u × v|
2
Area of the triangle with sides u and v
1
A4 = |u × v|
2
3
Volume of the parallelepiped with edges u, v and w
ax ay az
VP = |u ◦ (v × w)| = | bx by bz |
cx cy cz
Data Engineering
Linear Algebra
16 / 24
Vectors
Applications
Geometrical applications of vector products
1
Area of the parallelogram with sides u and v
AP = |u × v|
2
Area of the triangle with sides u and v
1
A4 = |u × v|
2
3
Volume of the parallelepiped with edges u, v and w
ax ay az
VP = |u ◦ (v × w)| = | bx by bz |
cx cy cz
4
Volume of a the tetrahedron with edges u, v and w
1
VT = |u ◦ (v × w)|
6
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Linear Algebra
16 / 24
Vectors
Applications
Lines and planes in R3
Data Engineering
Linear Algebra
17 / 24
Line is 3D space
Equations of lines
Equation of a line
An equation of a line passing through the point P0 (x0 , y0 , z0 ) in the
−
direction of the vector →
v = [a, b, c] is
[x, y, z] = [x0 , y0 , z0 ] + t [a, b, c] ,
Data Engineering
Linear Algebra
−
−
−
lub →
r =→
r0 + t→
v
18 / 24
Line is 3D space
Equations of lines
Equation of a line
An equation of a line passing through the point P0 (x0 , y0 , z0 ) in the
−
direction of the vector →
v = [a, b, c] is
[x, y, z] = [x0 , y0 , z0 ] + t [a, b, c] ,
Data Engineering
Linear Algebra
−
−
−
lub →
r =→
r0 + t→
v
18 / 24
Line is 3D space
Equations of lines
Various forms of the equation of the line l
vector equation:
[x, y, z] = [x0 , y0 , z0 ] + t [a, b, c]


x = x0 + at
, t∈R
parametric equations:
y = y0 + bt


z = z0 + ct
x − x0
y − y0
z − z0
symmetric equation:
=
=
a
b
c
Data Engineering
Linear Algebra
19 / 24
Plane in space
Definition
The plane
Definition
Given a fixed point P0 (x0 , y0 , z0 ) and a fixed vector N, the set of
−−→
points P (x, y, z) in R3 for which the vector P0 P is orthogonal
(perpendicular) to N is called a plane
Three points (not colinear) determine a plane in R3 .
Data Engineering
Linear Algebra
20 / 24
Plane in space
Equations of planes
Equation of a plane
−−→
vector P0 P = [x − x0 , y − y0 , z − z0 ] is orthogonal to N = [A, B, C]
Data Engineering
Linear Algebra
21 / 24
Plane in space
Equations of planes
Equation of a plane
−−→
vector P0 P = [x − x0 , y − y0 , z − z0 ] is orthogonal to N = [A, B, C]
−−→
P0 P ◦ N = 0
Data Engineering
Linear Algebra
21 / 24
Plane in space
Equations of planes
Equation of a plane
−−→
vector P0 P = [x − x0 , y − y0 , z − z0 ] is orthogonal to N = [A, B, C]
−−→
P0 P ◦ N = 0
[x − x0 , y − y0 , z − z0 ] ◦ [A, B, C] = 0
Data Engineering
Linear Algebra
21 / 24
Plane in space
Equations of planes
Equation of a plane
−−→
vector P0 P = [x − x0 , y − y0 , z − z0 ] is orthogonal to N = [A, B, C]
−−→
P0 P ◦ N = 0
[x − x0 , y − y0 , z − z0 ] ◦ [A, B, C] = 0
Equation of the plane
The plane π passing through the point P0 = (x0 , y0 , z0 ) with a
normal vector N = [A, B, C] is given by the equation
π:
A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0
π:
Data Engineering
Ax + By + Cz + D = 0
Linear Algebra
21 / 24
Plane in space
Equations of planes
Various form of the equation of the plane π
Normal form:
Intercept form:
Parametric form:
Data Engineering
Ax + By + Cz + D = 0
y
x
z
a + b + c =1
r = r0 + su + tv, s, t ∈ R


x = x0 + sa1 + ta2
, t, s ∈ R
y = y0 + sb1 + tb2


z = z0 + sc1 + tc2
Linear Algebra
22 / 24
Plane in space
Parallel and orthogonal planes
The normal vectors of distinct planes tell us about the relative
orientation of the planes.
Data Engineering
Linear Algebra
23 / 24
Plane in space
Parallel and orthogonal planes
The normal vectors of distinct planes tell us about the relative
orientation of the planes.
Parallel and Orthogonal planes
π1 ||π2
⇐⇒
π1 ⊥ π2
Data Engineering
n1 || n2 ,
⇐⇒
that is n1 = k · n2 ,
n1 ⊥ n2 ,
Linear Algebra
k∈R
that is n1 ◦ n2 = 0
23 / 24
Plane in space
Distances
Distances
The distance between the point P0 and the plane
π : Ax + By + Cz + D = 0
d(P0 , π) =
|Ax0 + By0 + Cz0 + D|
√
A2 + B 2 + C 2
The distance between two planes
d(π1 , π2 ) = √
Data Engineering
|D1 − D2 |
A2 + B 2 + C 2
Linear Algebra
24 / 24