Here in this post we will go through a quick recap of vector algebra keeping in mind that reader already had detail knowledge and problem solving skills related to the topic being discussed. Here we are briefing Vector Algebra because concepts of electrostatics , electromagnetism and many more physical phenomenon can best be conveniently expressed using this tool.

A vector is a quantity that requires both a magnitude (= 0) and a direction in space it can be represented by an arrow in space for example electrostatic force, electrostatic field etc. In symbolic form we will represent vectors by bold letters. In component form vector A is written as

Two vectors A and B can be added together to give another resultant vector C.

Two vectors A and B can be subtracted to give another resultant vector D.

When we multiply any vector A with any scalar quantity 'n' then it's direction remains unchanged and magnitude gets multiplied by 'n'. Thus,

n(

Scalar multiplication of vectors is distributive i.e.,

n(

Dot product of two vectors A and B is defined as the product of the magnitudes of vectors A and B and the cosine of the angle between them when both te vectors are placed tail to tail. Dot product is represented as

where θ is the angle between two vectors.

Result of dot product of two vectors is a scalar quantity.

Dot product is commutative :

Dot product is distributive :

Cross product or vector product of two vectors A and B is defined as

where nˆ is the unit vector pointing in the direction perpandicular to the plane of both A and B. Result of vector product is also a vector quantity.

Cross product is distributive i.e.,

In component form addition of two vectors is

where,

A = (A

Thus in component form resultant vector

C

C

C

In component form subtraction of two vectors is

where,

A = (A

Thus in component form resultant vector

D

D

D

NOTE:- Two vectors add or subtract like components.

= A

Thus for calculating the dot product of two vectors, first multiply like components, and then add.

= (A

Cross product of two vectors is itself a vector.

To calculate the cross product, form the determinantwhose first row is x, y, z, whose second row is A (in component form), and whose third row is B.

Vector product of two vectors can be made to undergo dot or cross product with any third vector.

(a) Scalar tripple product:-

For three vectors A, B, and C, their scalar triple product is defined as

obtained in cyclic permutation. If

(b) Vector Triple Product:-

For vectors A, B, and C, we define the vector tiple product as

Note that

(A . B)C ≠ A(B . C)

but

A vector is a quantity that requires both a magnitude (= 0) and a direction in space it can be represented by an arrow in space for example electrostatic force, electrostatic field etc. In symbolic form we will represent vectors by bold letters. In component form vector A is written as

**A**= A_{x}**i**+ A_{y}**j**+A_{z}**k****ADDITION OF VECTORS**Two vectors A and B can be added together to give another resultant vector C.

**C**=**A**+**B****SUBTRACTION OF VECTORS**Two vectors A and B can be subtracted to give another resultant vector D.

**D**=**A**-**B**=**A**+ (-**B**)**SCALAR MULTIPLICATION OF VECTOR**When we multiply any vector A with any scalar quantity 'n' then it's direction remains unchanged and magnitude gets multiplied by 'n'. Thus,

n(

**A**) = n**A**Scalar multiplication of vectors is distributive i.e.,

n(

**A**+**B**) = n**A**+n**B****DOT PRODUCT OF VECTORS**Dot product of two vectors A and B is defined as the product of the magnitudes of vectors A and B and the cosine of the angle between them when both te vectors are placed tail to tail. Dot product is represented as

**A**.**B**thus,**A**.**B**= |A| |B| cosθwhere θ is the angle between two vectors.

Result of dot product of two vectors is a scalar quantity.

Dot product is commutative :

**A.B**=**B.A**Dot product is distributive :

**A**. (**B**+**C**) =**A**.**B**+**A**.**C**also**A.A**= |A|^{2}**CROSS PRODUCT OF TWO VECTORS**Cross product or vector product of two vectors A and B is defined as

**A**x**B**= |A| |B| sinθ nˆwhere nˆ is the unit vector pointing in the direction perpandicular to the plane of both A and B. Result of vector product is also a vector quantity.

Cross product is distributive i.e.,

**A**x (**B**+**C**) = (**A**x**B**) + (**A**x**C**) but not commutative and the cross product of two parallel vectors is zero.**VECTOR ADDITION**In component form addition of two vectors is

**C**= (A_{x}+ B_{x})**i**+ (A_{y}+ B_{y})**j**+ (A_{y}+ B_{y})**k**where,

A = (A

_{x}, A_{y}, A_{z}) and B = (B_{x}, B_{y}, B_{z})Thus in component form resultant vector

**C**becomes,C

_{x}= A_{x}+ B_{x}C

_{y}= A_{y}+ B_{y}C

_{z}= A_{z}+ B_{z}**SUBTRACTION OF TWO VECTORS**In component form subtraction of two vectors is

**D**= (A_{x}- B_{x})**i**+ (A_{y}- B_{y})**j**+ (A_{y}- B_{y})**k**where,

A = (A

_{x}, A_{y}, A_{z}) and B = (B_{x}, B_{y}, B_{z})Thus in component form resultant vector

**D**becomes,D

_{x}= A_{x}- B_{x}D

_{y}= A_{y}- B_{y}D

_{z}= A_{z}- B_{z}NOTE:- Two vectors add or subtract like components.

**DOT PRODUCT OF TWO VECTORS****A.B**= (A_{x}i + A_{y}j + A_{z}k) . (B_{x}i + B_{y}j + B_{z}k)= A

_{x}B_{x}+ A_{y}B_{y}+ A_{z}B_{z}.Thus for calculating the dot product of two vectors, first multiply like components, and then add.

**CROSS PRODUCT OF TWO VECTORS****A x B**= (A_{x}**i**+ A_{y}**j**+ A_{z}k) x (B_{x}i + B_{y}j + B_{z}**k**)= (A

_{y}B_{z}- A_{z}B_{y})**i**+ (A_{z}B_{x}- A_{x}B_{z})**j**+ ( A_{x}B_{y}- A_{y}B_{x})**k**.Cross product of two vectors is itself a vector.

To calculate the cross product, form the determinantwhose first row is x, y, z, whose second row is A (in component form), and whose third row is B.

**VECTOR TRIPPLE PRODUCT**Vector product of two vectors can be made to undergo dot or cross product with any third vector.

(a) Scalar tripple product:-

For three vectors A, B, and C, their scalar triple product is defined as

**A . (B x C) = B . (C x A) = C . (A x B)**obtained in cyclic permutation. If

**A**= (A_{x}, A_{y}, A_{z}) ,**B**= (B_{x}, B_{y}, B_{z}) , and**C**= (C_{x}, C_{y}, C_{z}) then**A . (B x C)**is the volume of a parallelepiped having A, B, and C as edges and can easily obtained by finding the determinant of the 3 x 3 matrix formed by**A**,**B**, and**C**.(b) Vector Triple Product:-

For vectors A, B, and C, we define the vector tiple product as

**A x (B x C) = B(A . C) - C(A - B)**Note that

(A . B)C ≠ A(B . C)

but

**(A . B)C = C(A . B).**
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