In this post we’ll lern Vector algebra in component form.

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 is not equal to A(B . C)

but

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

*Download complete vector algebra synopsis as pdf*