Vectors Revision in One Shot

Introduction

• Special video for NEET/JEE aspirants to revise the entire chapter on vectors.
• This session will cover important concepts and formulas of vectors.
• Vectors do not standalone questions but are important in other chapter questions.

Basics of Vectors

• A vector is defined by its magnitude and direction.
• Denoted by a vector (â⃗), magnitude (|a|) gives its size, and direction is its orientation.

Types of Vectors

• Equal, parallel, anti-parallel, and zero vectors are common types.

Addition of Vectors

• Several methods include head-tail method:
  • Align the tail of one vector to the head of another.
  • Join the head of the last vector to the tail of the first vector.
• Resultant (R) is drawn from the start to the end point after joining.
• Parallelogram Law of Addition:
  • Two vectors a and b form adjacent sides of a parallelogram.
  • Resultant (R) is given by the diagonal passing through the common point of the vectors.
  • Formula: R = sqrt[a² + b² + 2ab * cosθ]
  • Direction, tanα = (bsinθ)/(a+bcosθ)
• Subtraction of Vectors (a-b):
  • Subtraction is the addition of one vector with the negative of another.
  • a - b = a + ( -b )*

Special Cases:

• When the sum and difference of two vectors are equal in magnitude, they are perpendicular.

Multiplication of Vectors

• Types: Dot and Cross product.
  • Dot Product: a.b = |a| |b| cosθ (results in scalar).
  • Cross Product: (a x b) results in a vector C, perpendicular to both a and b.
  • Formula: |a x b| = |a||b|sinθ
  • Direction Determination with Right-Hand Rule or corkscrew rule.

Special Cases:

• Perpendicular Vectors (a . b = 0).
• Parallel Vectors (a x b = 0).

Cartesian Form of Vectors i,j,k

• Unit vectors along x,y,z axes: i, j, k respectively.
• Any vector a = ai + bj + ck.

Important Formulas:

• ∣a∣ = sqrt((a_x)² +(a_y)² +(a_z)² )
• Unit vector in the direction of a: a^ = a / ∣a∣

Calculation in Cartesian Form

• Addition/Subtraction:
  • A = 3i + 2j + k and B = 2i - j + 3k.
  • A + B = (3+2)i + (2-1)j + (1+3)k = 5i + j + 4k.
  • A - B: (3-2)i + (2+1)j + (1-3)k = i + 3j - 2k.
• Dot Product in Cartesian Form:
  • A⋅B = AxBx + AyBy + AzBz (scalar).
• Cross Product in Cartesian Form:
  • A × B calculated using determinant method involving i,j,k.
• AxB (cross-product determination method) [Example detailed in content given.]

Resolution of Vectors

• Resolve a vector into x and y components.
  • Avector = A cosθ along x and A sinθ along y.

Three Vectors Addition

• Breakdown each vector into x and y components.
  • Rx = Ax + Bx + Cx
  • Ry = Ay + By + Cy
  • R = sqrt(Rx² + Ry²)

Example Problems