Component of any vector is the projection of that vector along the three coordinate axis X, Y, Z.

**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).**