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 =
Degree
Radian
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 = ±AB
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 (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 obey 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 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: