Operations on Vectors
Vector definition
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 Addition Vectors Dot Product
Vectors Cross Product Vectors functions and derivation Integration of Vectors
Vector spherical cylindrical coordinates
Vectors Print ellipse
Vectors definition

A vector V is represented in three dimentional 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 propotional when each vector is a scalar multiple of the other and neither is zero.
Vectors addition (A ± B)
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)
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 AB 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 obey the following laws:
The commutative law does not hold for cross product because:
AB = − (BA)         and         A ✕ (BC) ≠ (AB) ✕ C
Distributive law: A ✕ (B + C) = AB + AC
(A + B) ✕ C = (AC) + (BC)
If m is a scalar then: m(AB) = (mA) ✕ B = A ✕ (mB) = (AB)m

Triple scalar product is defined as the determinant:
(AB)C = -C(AB) = C(BA)

Other relations: A · (AC) = 0
A(BC) + B(CA) + C(AB) = 0
A(BC) =(A · C)B − (A · B)C
(AB)C = (A · C)B(B · C)A
(AB) · (CD) = (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 presentation 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 is dotted into a vector V,the result is the divergence of V.
Note that:
Curl When the vector operator is crossed into a vector V,the result is the curl of V.
Laplacian the dot product of the vector into itself gives the scalar operator known as Laplacian operator.
Laplacian operator
Laplace's equation
Other relations of the operator.
ᐁ ✕ (ᐁΦ ) = 0
ᐁ· (ᐁ✕A ) = 0
ᐁ✕(ᐁ✕A ) = ᐁ(ᐁ·A)−ᐁ2A
ᐁ(Φ + ψ ) = ᐁΦ + ᐁψ
ᐁ·(A + B ) = ᐁ·A + ᐁ·B
ᐁ✕(A + B ) = ᐁ✕A + ᐁ✕B
ᐁ·(ΦA ) = (ᐁΦA + Φ(ᐁ·A )
ᐁ✕(ΦA) = (ᐁΦ)✕A + Φ(ᐁ✕A )
ᐁ· (AB ) = B · (ᐁ✕A ) − A·(ᐁ✕B )
ᐁ✕(AB ) = (B ·ᐁ)AB(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
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
Vectors spherical coordinate
Spherical system of coordinates are 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 coordinats:
Transformation from spherical to cartesian coordinats:
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 coordinats:
Transformation from spherical to cartesian coordinats:
x = r cos θ
y = r sin θ
z = z