Scalars and Vectors

• Scalars are physical quantities described by magnitude and units, e.g., mass, length, time, density, energy, work, temperature, and charge.
• Scalars are added, subtracted, and multiplied using ordinary algebra.
• Vectors are physical quantities described by magnitude, unit, and direction.
• Examples of vectors are force, velocity, acceleration, momentum, torque, electric field, and magnetic field.
• Vectors are added, subtracted, and multiplied using vector algebra.

Vector Representation

• Vectors are represented symbolically using bold letters (F, a, d) or with a bar or arrow over the symbol ( ̅ ̅ ̅ ⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗).
• Graphically, vectors are represented as a line segment with an arrowhead.
  • Line ⃗⃗⃗⃗⃗ with arrowhead at B represents vector where the length of AB gives the magnitude of , and the direction from A to B gives the direction of .

Position Vector:

• Position vectors have an initial point at the origin O and a terminal point at P, written as ⃗⃗⃗⃗⃗.

Vector Representation in Coordinate Systems:

• In a 2D Cartesian plane = { (x,y) : x,y ∈ R}, a vector is written as ⃗⃗⃗⃗⃗ = x i+ yj
• In a 3D coordinate system = { (x,y,z) : x,y,z ∈ R}, a vector is written as ⃗⃗⃗⃗⃗ = x i+ yj + zk

Magnitude (Length or Norm):

• The magnitude (length or norm) of vector ⃗⃗⃗⃗⃗ is its absolute value, denoted as |⃗⃗⃗⃗⃗ |.
• In 3D, magnitude is calculated as |⃗⃗⃗⃗⃗ |=√

Null or Zero Vector:

• A null or zero vector has zero magnitude.

Unit Vector:

• A unit vector has unit magnitude and the same direction as the given vector, represented as ̂ , ̂ , ̂, ̂ , ̂ , ̂.
• The unit vector of vector ⃗ is  =⃗⃗/|⃗⃗ |.

Direction Cosines:

• For ⃗⃗ = Ax ̂ +Ay ĵ+ Az k, if ⃗ makes angles α, β, and γ with the x, y, and z-axes, then direction cosines are defined as:
  • Cos α =Ax/|⃗⃗ |, Cos β = Ay/|⃗⃗ |, Cos γ = Az/|⃗⃗ |

Vector Addition:

• Vector addition combines two or more vectors into a single vector.
• Vector addition uses the graphical Head To Tail Rule.

Resultant Vector:

• The resultant vector is the sum of two or more vectors.

Rectangular Components:

• Rectangular components of a vector are perpendicular to each other.

Collinear Vectors:

• Vectors ⃗ and ⃗ are collinear if ⃗ = λ⃗ where λ is a scalar.
  • If 𝜆 > 0, ⃗ and ⃗ are parallel vectors.
  • If 𝜆 < 0, ⃗ and ⃗ are anti-parallel vectors.
  • If 𝜆 = 0, ⃗ and ⃗ are equal vectors (⃗ = ⃗).

Free Vectors:

• Free vectors have positions that are not fixed in space.
• Example: displacement.

Localized Vectors:

• Localized vectors cannot be shifted parallel to themselves, and their line of action is fixed.
• Examples: force and momentum.

Parallel Vectors:

• Parallel vectors are two or more vectors having the same direction.
  • If =ax i+ay j+az k and b=bx i+by j+bz k are parallel, their directional components are proportional: ax/bx = ay/by = az/bz.

Perpendicular Vectors:

• Perpendicular vectors are two or more vectors making an angle of 90° with each other.
  • If ⃗ =ax i+ay j+az k and b=bx i+by j+bz k are perpendicular then the sum of the product of their directional components is zero: ax bx+ay by+az bz = 0.

Properties of Vector Addition

• Commutative Property: ⃗ + ⃗ = ⃗ + ⃗
• Associative Property: (⃗ + ⃗ ) + = ⃗ + (⃗ + )
• Scalar Multiplication: λ⃗ is scalar multiplication with vector a.
  • (λ + µ) ⃗ = λ ⃗ + µ ⃗
  • λ ( + ⃗ )= λ ⃗ + λ ⃗

Coplanar Vectors Theorem:

• If , , and are three given non-coplanar vectors, then any vector can be expressed uniquely as a linear combination if , , i.e., = x + y+z where x, y, and z are scalars.

Position Vector Division Theorem:

• The position vector of a point P which divides the join of two given points A and B, with position vectors and in the ratio λ : µ, is given by

μ𝑎+λ 𝑏/μ +λ

• Special Case: if 𝜆 = µ, then P is the midpoint of AB and if = ⃗ + ⃗ /2