Vectors

Vectors:

It is defined as;

Quantity having magnitude and direction are  called vectors and they are represented by arrow over the letters.

 Types of Vectors:

Types of vectors are as follows:

1. Free Vectors:

Vector having magnitude and direction and can be applied at any point in the space is called Free vector.

2. Sliding Vectors:

Thsee vectors have a line of action in space but no particular point if application 

3. Fixed Vectors:

Vectors having specific point of application are called fixed vectors.

Pallelogram Law of Vector Addition:

If two vectors ate considered to be the adjacent sides of a parallelogram, then the resultant of two vectors is given by drawing the diagonal between the vectors.

The formula used for the Parallelogram Law of Vector Addition is:

Resolution of Vectors:

It is defined as;

 The converting of a vector into its rectangular components.

1. For 2-dimensional Space:

Now the rectangular componets of vector a are:

ax = a cosθi

ay = a sinθj

And, the rectangular components of vector b are:

bx = b cosαi

by = b sinαj

Now, the resultant X and Y comoments are:

Rx = a cosθi + b cosαi

Ry = a sinθj + b sinαj

And, the resultant vector and angle is given by:

R = √ Rx² + Ry²

ɸ = tan⁻¹ ( Rx / Ry )

2. For 3-dimensional Space:

Where, 

• Ax, Ay, Az are the projections of vector A
• α, β, ɣ are direction cosines

Now, the vector form of vector A is given b:

Vector A = Axi + Ayj + Azk 

The magnitude of resultant vector A is given by:

A = √ Ax² + Ay² + Az²

The respective angles are given by:

cosα = vector Ax / A

cosβ = vector Ay / A

cosɣ = vector Az / A

The unit vector is given by:

Unit vector Ua = cosα + cosβ + cosɣ

Or 

Vector A = A × (unit vector Ua)

Vector A = Acosα + Acosβ + Acosɣ

And, 

A = √ Ax² + Ay² + Az²

A = √ (Acosα)² + (Acosβ)² + (Acosɣ)²

A = A √ (cosα)² + (cosβ)² + (cosɣ)²

cosα² + cosβ² + cosɣ² = 1

Position Vector:

It is defined as;

Vector which specifies the position of point in space is called Position Vector.

Dot Product:

It is defined as;

The scalar product between two vectors.

References:

• www.youtube.com
• www.wikipedia.com
• http://web.aeromech.usyd.edu.au/statics/doc/friction/Friction1.htm
• From Book Engineering Mechanics statics and dynamics by R.C. Hibbeler