Operations on Vectors
A = i + j + k
B = i + j + k

|A|
Unit Vector:
Angle to x axes:
Angle to y axes:
Angle to z axes:
Spherical Coordinate:
r = θ = Φ =
|B|
Unit Vector:
Angle to x axes:
Angle to y axes:
Angle to z axes:
Spherical Coordinate:
r = θ = Φ =









|C|
Angle Between Vectors:
Radian = Degree =

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 of a rectangle defined by the vectors A = 4i and B = 3j and the unit vector in R direction, also find angle θ by the dot product.

Diagonal vector:
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)


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.
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 (AB)


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 A(1, -2, 3),
B(3, 1, 2) and C(2, 3, -1).

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: