Operations on Vectors
▲
Vector
A =
i +
j +
k
|A|
Unit Vector:
Angle to axis:
x =
y =
z =
Spherical Coordinate:
r =
θ =
Φ =
Vector
B =
i +
j +
k
|B|
Unit Vector:
Angle to axis:
x =
y =
z =
Spherical Coordinate:
r =
θ =
Φ =
Degree
|C|
Angle Between Vectors:
Vectors Definition
Vectors Cross Product
Vectors functions and derivation
Vectors Addition
Integration of Vectors
Vector spherical cylindrical coordinates
Vectors Dot Product
Vectors
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Vectors definition
A vector V is represented in three-dimensional space in terms of the sum of its three mutually perpendicular components.
Where i, j and k are the unit vector in the x, y and z directions respectively and has magnitude of one unit.
The scalar magnitude of V is:
Let V be any vector except the 0 vector, the unit vector q in the direction of V is defined by:
A set of vectors for example {u, v, w} is linearly independent if and only if the determinant D of the vectors is not 0.
Two vectors V and Q are said to be parallel or proportional, when each vector is a scalar multiple of the other and neither is zero.
Vectors addition (A ± B)
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Two vectors A and B may be added to obtain their resultant or sum A + B, where the two vectors are the two legs of the parallelogram.
Vectors addition obey the following laws:
Commutative law:
A + B = B + A
Associative law:
A + (B + C) = (A + B) + C
(c and d are any number)
c(dA) = (cd)A
Distributive law:
(c + d)A = cA + dA
c(A + B) = cA + cB
The subtraction of a vector is the same as the addition of a negative vector
A − B = A + (−B)
1 · A = A
0 · A = 0
(− 1)A = −A
If A and B are two vectors then the following relations are true:
Example:
find the diagonal length and the unit vector of a rectangle defined by the vectors A = 4i and B = 3j in the R direction, also find angle θ by the dot product.
Solution:
Diagonal vector:
R = A + B = 4i + 3j
Diagonal length:
The unit vector in the R direction is:
dot product
A · R in order to find cosθ:
Dot or scalar product (A · B)
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Dot or scalar product (A · B)
The dot or scalar product of two vectors A and B is defined as:
(The result is a scalar value)
θ is the angle between the two vectors
From the definition of the dot product, it follows that:
Thus:
And
Two vectors are perpendicular when their dot product is:
A · B
= 0
The angle θ between two vectors A and B is:
Where l, m and n stands for the respective direction cosines of the vectors. It is also observed that two vectors are perpendicular when their direction cosines obey the relation:
Where:
The dot product satisfies the following laws:
Commutative law:
A · B = B · A
Distributive law:
A · (B + C) = A · B + A · C
If m is a scalar:
m(A · B) = (mA) · B = A · (mB) = (A · B)m
Other relations:
(A · B)C ≠ A(B · C)
A · A = |A|
2
Parallel vectors when:
A · B = ±|A||B|
Example:
given two vectors A = 2i + 2j − k and B = 6i − 3j + 2k. Find the vectors dot product and the angle between the vectors.
Solution
The dot product is:
A · B
= 2 · 6 − 2 · 3 − 1 · 2 = 4
Magnitude of A and B are:
The angle between the vectors is:
Cross or vector product (A✕ B)
▲
The cross or vector product of two vectors A and B is defined as:
(The result is a vector)
n - unit vector whose direction is perpendicular to vectors A and B.
Note:
the direction of
A
✕
B
is normal to the plane defined by
A
and
B
and is pointing according to the right-hand screw rule.
From the definition of the cross product the following relations between the vectors are apparent:
The vector product is written as:
This expression may be written as a determinant:
The cross product obeys the following laws:
The commutative law does not hold
for cross product because:
A
✕
B
= −
(B
✕
A)
and
A
✕ (
B
✕
C
) ≠ (
A
✕
B
) ✕ C
Distributive law:
A
✕ (
B
+
C
) =
A
✕
B
+
A
✕
C
(
A
+
B
) ✕
C
= (
A
✕
C
) + (
B
✕
C
)
If m is a scalar, then:
m(
A
✕
B
) = (m
A
) ✕
B
= A ✕ (m
B
) = (
A
✕
B
)m
Triple scalar product is defined as the determinant:
(A
✕
B)
✕
C
= -
C
✕
(A
✕
B)
=
C
✕
(B
✕
A)
Other relations:
A
· (
A
✕
C
) = 0
A
✕
(B
✕
C)
+
B
✕
(C
✕
A)
+
C
✕
(A
✕
B)
= 0
A
✕
(B
✕
C) =(A
·
C)B − (A
·
B)C
(A
✕
B)
✕
C
=
(A
·
C)
B
−
(B
·
C)
A
(A
✕
B)
·
(C
✕
D)
=
(A
·
C)
(B
·
D)
−
(A
·
D)
(B
·
C)
Example:
find the area of the triangle ABC and the equation of the plane passing through points A, B and C if the points coordinates are: A(1, -2, 3), B(3, 1, 2) and C(2, 3, -1).
Solution:
the vectors presentations are:
Vector
Vector
The area of the parallelogram is:
The area of triangle ABC is:
The normal to the plane vector is:
The plane equation will be accordingly:
Vectors functions and derivation
▲
The derivative of a vector P according to a scalar variable t is:
The derivative of the sum of two vectors is:
The derivative of the product of a vector P and a scalar u(t)according to t is:
The derivative of two vectors dot product:
For the cross product the derivative is:
Gradient, If ϕ is a scalar function defined by ϕ=f(x,y,z), we define the gradient of ϕ, that is a vector in the n-direction and represents the maximum space rate of change of ϕ.
The del operator:
Gradient operator:
divergence, when the vector operator
ᐁ
is dotted into a vector V, the result is the divergence of V.
Note that:
Curl, When the vector operator
ᐁ
is crossed into a vector V, the result is the curl of V.
Laplacian the dot product of the vector
ᐁ
into itself gives the scalar operator known as Laplacian operator.
Laplacian operator
Laplace's equation
Other relations of the
ᐁ
operator.
ᐁ ✕ (ᐁ
Φ
) = 0
ᐁ· (ᐁ✕
A
) = 0
ᐁ✕(ᐁ✕
A
) = ᐁ(ᐁ·
A
)−ᐁ
2
A
ᐁ(
Φ
+
ψ
) = ᐁ
Φ
+ ᐁ
ψ
ᐁ·(
A
+
B
) = ᐁ·
A
+ ᐁ·
B
ᐁ✕(
A
+
B
) = ᐁ✕
A
+ ᐁ✕
B
ᐁ·(
ΦA
) = (ᐁ
Φ
)·
A
+
Φ
(ᐁ·
A
)
ᐁ✕(
ΦA
) = (ᐁ
Φ
)✕
A
+
Φ
(ᐁ✕
A
)
ᐁ· (
A
✕
B
) =
B
· (ᐁ✕
A
) −
A
·(ᐁ✕
B
)
ᐁ✕(
A
✕
B
) = (
B
·ᐁ)
A
−
B
(ᐁ·
A
) − (
A
·ᐁ)
B
+
A
(ᐁ·
B
)
ᐁ(
A
·
B
) = (
B
·ᐁ)
A
+ (
A
·ᐁ)
B
+
B
✕(ᐁ✕
A
)
A
✕(ᐁ✕
B
)
Vectors integration
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Vectors integration
If V is a function of x, y, and z and an element of volume is dv = dx dy dz, the integral of V over the volume may be written as the vector sum of the three integrals of its components:
Vector Spherical and Cylindrical Coordinates
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Vectors spherical coordinate
Spherical system of coordinate is defined by (r, θ, ϕ)
r
− Distance of the point P from the origin
θ
− Is the angle from the x-z plane
ϕ
− Is the angle from the z axis to the point P
Transformation from cartesian to spherical coordinates:
Transformation from spherical to cartesian coordinates:
x = r
sin
Φ
cos
θ
y = r
sin
Φ
sin
θ
z = r
cos
Φ
Vectors cylindrical coordinate
Cylindrical system of coordinates are defined by (r, θ, z)
r −
Distance of the point P from the origin
θ −
Is the angle from the x-z plane
z −
Is the same as in cartesian coordinate
Transformation from cartesian to spherical coordinates:
Transformation from spherical to cartesian coordinates:
x = r
cos
θ
y = r
sin
θ
z = z