Vector Multiplication - Dot & Cross Product - Field Theory.

Vector Multiplication:


When two vectors are multiplied the result is either a scalar or a vector depending on how they are multiplied.


The two important types of vector multiplication are:

• Dot Product/Scalar Product (A.B)
• Cross product (A x B)

DOT PRODUCT (A . B):


- Dot product of two vectors A and B is given as:

A . B = |A| |B| cosθ_AB

Where θ_AB is the angle formed between A and B.
Also θ ranges from 0 to π i.e. 0 ≤ θ_AB ≤ π


- The result of A.B is a scalar, hence dot product is also known as Scalar Product.


- If A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z) then A.B = A_xB_x + A_yB_y + A_zB_z


- If A.B = |A| |B|, then obviously cosθ_AB =1 which means θ_AB = 0^o
This shows that A and B are in the same direction or we can also say that A and B are parallel to each other.


- If A.B = - |A| |B|, then obviously cosθ_AB = -1 which means θ_AB = 180^o.
This shows that A and B are in the opposite direction or we can also say that A and B are antiparallel to each other.


- Similarly if A.B = 0, then cosθ_AB =0 which means θ_AB =90^o.
This shows that A and B are orthogonal or perpendicular to each other.


- Since we know the Cartesian base vectors are mutually perpendicular to each other, we have

a_x . a_x = a_y . a_y = a_z . a_z = 1

a_x . a_y = a_y . a_z = a_z . a_x = 0

CROSS PRODUCT ( A x B):

- Cross Product of two vectors A and B is given as:

A x B = |A| |B| sinθ_AB a_n

Where θ_AB is the angle formed between A and B and a_n is a unit vector normal to both A and B.
Also θ ranges from 0 to π i.e. 0 ≤ θ_AB ≤ π


- The cross product is an operation between two vectors and the output is also a vector.
If A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z) then,

- The resultant vector is always normal to both the vector A and B.


- If A x B = 0, then sin θ_AB = 0 which means θ_AB = 0^o or 180^o;
This shows that A and B are either parallel or antiparallel to each other.


- Since we know the Cartesian base vectors are mutually perpendicular to each other, we have

 a_x x a_x = a_y x a_y = a_z x a_z =0

a_x x a_y = a_z

a_y x a_z = a_x

a_z x a_x = a_y

SCALAR TRIPLE PRODUCT:

- A . (B x C) = B . (C x A) = C . (A x B)

- Volume of a parallelogram having A, B and C as edges is given by the Scalar Triple Product.


VECTOR TRIPLE PRODUCT:


- A x (B x C) = B (A . C) - C (A .B)


COMPONENT OF A VECTOR:


- Scalar Component A_B of vector A along vector B

A_B = Acosθ_AB = A |a_B|cosθ_AB = A.a_B

- Vector Component A_B of vector A along vector B
is the scalar product multiplied by the unit vector along B
i.e.  A_B = (A.a_B) a_B



To Prove : (A * B) . A = 0.

A * B = a_x ( A_y B_z - A_z B_y ) -  a_y ( A_x B_z - A_z B_x ) +  a_z ( A_x B_y - A_y B_x )

A = A_x a_x + A_y a_y + A_z a_z

(A * B ) . A = A_x A_y B_z - A_x A_z B_y - A_x A_y B_z + A_y A_z B_x + A_x A_z B_y - A_y A_z B_x     =    0.



SOLVED EXAMPLES / NUMERICALS:

Q.1. Given vectors A = a_x + 3a_z and B = 5a_x + 2a_y -6a_z, determine

a) |A+B|
B) 5A – B
c) Component of A along a_y.
d) Unit vector parallel to 3A + B.              SOLUTION/ANSWER


Q.2 Given Points P (1, -3, 5), Q (2, 4, 6) and R (0, 3, 8) find
a) Position vectors of P and R.
b) Distance vector r_QR.
c) Distance between Q and R.                   SOLUTION/ANSWER


Q. 3 Find the unit vector along the line joining point (2, 4, 4) to point (-3, 2, 2) ?               SOLUTION/ANSWER


Q. 4 Given that A = 3a_x + 5a_y – 7a_z and B = a_x – 2a_y + a_z ; find
a) | 2B + 0.4A |
b) A.B - | B |²
c) A x B                                   SOLUTION/ANSWER



ALSO READ:

- Scalars and Vectors.

- Unit Vectors.

- Position and Distance Vectors.

- Vector Multiplication.

- Components Of a Vector.

- Line , Surface and Volume Intergral.

- Del Operator - Definition and Significance.

- Gradient Of a Scalar (∇ V).

- Divergence Of a Vector ( ∇ . A ).

- Curl Of a Vector ( ∇ x A).

- Laplacian Of a Scalar ( ∇² V).


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