Comprehensive Vector Analysis: Definitions, Coordinate Geometry, Addition Rules, and Multiplication

Fundamental Concepts of Vectors

  • Definition of Vectors:

    • Physical quantities that require magnitude with a proper unit along with direction are defined as vectors.
    • Position vectors are also included under vector quantities.
    • Examples: Acceleration, torque, force, etc.
    • Specific Example: A force vector of magnitude 30 N30\,\text{N} directed towards North (F⃗=30 N North\vec{F} = 30\,\text{N}\,\text{North}).

  • Negative Vector:

    • Definition: A vector whose magnitude is equal to another vector but acts in the opposite direction to that vector.
    • Mathematical Representation: A⃗=−A⃗\vec{A} = -\vec{A} or B⃗+(−B⃗)=0\vec{B} + (-\vec{B}) = 0
    • Example: If a vector A⃗\vec{A} has a magnitude of 3 m3\,\text{m} pointing to the right, its negative vector −A⃗-\vec{A} has a magnitude of 3 m3\,\text{m} pointing to the left (represented as −3 m-3\,\text{m}). Adding them together yields: 3+(−3)=03 + (-3) = 0

  • Null Vector (Zero Vector):

    • Definition: If two equal and opposite direction vectors are added, their resultant is a zero magnitude vector with an arbitrary (random) direction.
    • Examples:
      • Null vector symbol: a⃗=0\vec{a} = 0
      • Velocity of a body at rest: v⃗=0\vec{v} = 0
      • Net force acting on an object under balanced forces: F⃗net=0\vec{F}_{\text{net}} = 0

  • Unit Vector:

    • Definition: A vector whose magnitude is one (11) and is used exclusively to represent direction.
    • Symbolic Representation: Denoted by a cap symbol over the vector letter, such as A^\hat{A} ("A cap").
    • Vector Structure: A full vector is expressed as the product of its scalar magnitude and its unit vector direction:         A⃗=AA^\vec{A} = A \hat{A}
    • Example: A⃗=10 m East\vec{A} = 10\,\text{m}\,\text{East}, where 10 m10\,\text{m} is the magnitude and East\text{East} is the direction.
    • Cartesian Coordinate System Unit Vectors:
      • Unit vector along the x-axisx\text{-axis}: i^\hat{i}
      • Unit vector along the y-axisy\text{-axis}: j^\hat{j}
      • Unit vector along the z-axisz\text{-axis}: k^\hat{k}
      • Spatial positions are specified using standard coordinates (x,y,z)(x, y, z).

  • Equal Vectors:

    • Definition: Vectors that possess both the exact same magnitude and the exact same direction.
    • Example: Given vector A⃗=2 N right\vec{A} = 2\,\text{N}\,\text{right} and vector B⃗=2 N right\vec{B} = 2\,\text{N}\,\text{right}, vector A⃗\vec{A} and vector B⃗\vec{B} are equal vectors because their magnitudes (2 N2\,\text{N}) and directions (right) are identical.

Vector Angles and Coordinate Geometry

  • Standard Rules for Angles:

    • In coordinate systems, angles are measured counterclockwise relative to specified axes.
    • When determining the angle formed between two vectors, the vectors must be joined tail-to-tail.

  • Angle Calculation Problems:

    • Problem 1: A force F=5 NF = 5\,\text{N} makes an angle of 45∘45^\circ with the −x-axis-x\text{-axis}. Find its angle with the +x-axis+x\text{-axis} and +y-axis+y\text{-axis}.

      • Angle with +x-axis+x\text{-axis}: Sweeping counterclockwise from the positive x-axisx\text{-axis} through Quadrant I (90∘90^\circ), Quadrant II (90∘90^\circ), and adding the 45∘45^\circ displacement into Quadrant III:             θ(+x)=90∘+90∘+45∘=225∘\theta_{(+x)} = 90^\circ + 90^\circ + 45^\circ = 225^\circ
      • Angle with +y-axis+y\text{-axis}: Sweeping counterclockwise from the positive y-axisy\text{-axis} through Quadrant II (90∘90^\circ) plus the 45∘45^\circ angle:             θ(+y)=90∘+45∘=135∘\theta_{(+y)} = 90^\circ + 45^\circ = 135^\circ

    • Problem 2: A vector lies in Quadrant IV making an angle of 30∘30^\circ with the +x-axis+x\text{-axis} (measured clockwise below the axis). Find its angle measured counterclockwise from the +x-axis+x\text{-axis}.

      • Angle with +x-axis+x\text{-axis}: Summing the quadrant sweeps counterclockwise through Quadrant I (90∘90^\circ), Quadrant II (90∘90^\circ), Quadrant III (90∘90^\circ), and Quadrant IV (60∘60^\circ):             θ(+x)=90∘+90∘+90∘+60∘=330∘\theta_{(+x)} = 90^\circ + 90^\circ + 90^\circ + 60^\circ = 330^\circ

    • Problem 3: A vector makes an angle of 45∘45^\circ with the −y-axis-y\text{-axis}. Find its angle with the +y-axis+y\text{-axis}.

      • Angle with +y-axis+y\text{-axis}: Measuring counterclockwise from the positive y-axisy\text{-axis} across the straight angle (180∘180^\circ) plus the 45∘45^\circ offset:             θ(+y)=180∘+45∘=225∘\theta_{(+y)} = 180^\circ + 45^\circ = 225^\circ

  • Angle Between Vector Sides of Geometric Shapes (Equilateral Triangle):

    • Question: What is the angle between two sides (vectors) of an equilateral triangle?
    • Answer: 120∘120^\circ
    • Explanation: The angle between two vectors is formed by joining their tails. In an equilateral triangle, adjacent sides placed head-to-tail form an internal angle of 60∘60^\circ. Joining the vectors tail-to-tail requires extending one vector along its line of action, yielding the exterior angle:         θ=180∘−60∘=120∘\theta = 180^\circ - 60^\circ = 120^\circ

Vector Addition and Subtraction Operations

  • Cases of Vector Addition:

    • Parallel Vectors (Same Direction):
      • Adding 3 N3\,\text{N} and 2 N2\,\text{N} in the same direction yields a resultant magnitude of:             3 N+2 N=5 N3\,\text{N} + 2\,\text{N} = 5\,\text{N}
    • Anti-parallel Vectors (Opposite Direction):
      • Adding 3 N3\,\text{N} in one direction and 2 N2\,\text{N} in the opposite direction yields:             3 N+(−2 N)=3 N−2 N=1 N3\,\text{N} + (-2\,\text{N}) = 3\,\text{N} - 2\,\text{N} = 1\,\text{N}
    • Vectors at an Angle:
      • Vectors oriented at arbitrary angles are added geometrically by placing the tail of the second vector at the head of the first vector.

  • Triangular Rule of Vector Addition:

    • Definition: If two vectors are represented by two sides of a triangle using head-to-tail rules, then their resultant vector is represented by the third side of the triangle, taken from the tail of the first vector to the head of the second vector.
    • Equation: R⃗=A⃗+B⃗\vec{R} = \vec{A} + \vec{B}

  • Commutative Law of Vector Addition:

    • Vector addition obeys the commutative law; changing the order of vectors does not change the resultant vector.
    • Mathematical Expression: A⃗+B⃗=B⃗+A⃗=R⃗\vec{A} + \vec{B} = \vec{B} + \vec{A} = \vec{R}

  • Parallelogram Rule of Vector Addition:

    • Definition: If two vectors acting at a point are represented by two adjacent sides of a parallelogram, then their resultant vector is represented by the diagonal of the parallelogram passing through that point.
    • Mathematical Expression: A⃗+B⃗=B⃗+A⃗=R⃗\vec{A} + \vec{B} = \vec{B} + \vec{A} = \vec{R}

  • Resultant of a Closed Figure:

    • Rule: The resultant of vectors forming a closed figure (polygon) connected head-to-tail is always zero.
    • Example: For three vectors forming a closed triangle loop:         A⃗+B⃗+C⃗=0\vec{A} + \vec{B} + \vec{C} = 0

  • Vector Subtraction:

    • Vector subtraction is defined as the addition of a negative vector.
    • Mathematical Formula: A⃗−B⃗=A⃗+(−B⃗)\vec{A} - \vec{B} = \vec{A} + (-\vec{B})
    • Graphical Method:
      1. Reverse the direction of vector B⃗\vec{B} by 180∘180^\circ to create the negative vector −B⃗-\vec{B}.
      2. Join the tail of −B⃗-\vec{B} to the head of vector A⃗\vec{A}.
      3. Draw the resultant vector R⃗\vec{R} from the tail of A⃗\vec{A} to the head of −B⃗-\vec{B}.

Multiplication of Vectors

  • Categories of Multiplication:

    • Vector multiplication is divided into two distinct processes:
      1. Multiplication of a scalar with a vector.
      2. Multiplication of a vector with a vector.

  • Multiplication of a Scalar with a Vector:

    • Multiplying a vector by a scalar number scales the magnitude of the vector by that number.
    • Positive Scalar: Multiplying by a positive scalar increases or decreases magnitude while preserving direction.
    • Negative Scalar: Multiplying by a negative scalar scales the magnitude and reverses the direction by 180∘180^\circ
    • Examples:
      • Given vector F⃗=2 N East\vec{F} = 2\,\text{N}\,\text{East}.
      • Multiplying by scalar 22:             2×F⃗=2×2 N East=4 N East2 \times \vec{F} = 2 \times 2\,\text{N}\,\text{East} = 4\,\text{N}\,\text{East}
      • Multiplying by scalar −2-2:             −2×F⃗=−2×2 N East=−4 N East=4 N West-2 \times \vec{F} = -2 \times 2\,\text{N}\,\text{East} = -4\,\text{N}\,\text{East} = 4\,\text{N}\,\text{West}

  • Multiplication of Vector with Vector:

    • Scalar Product (Dot Product):
      • Definition: Occurs when multiplying two vectors results in a scalar product.
      • Example: Work done, defined as the product of force and displacement:             Work=Force×displacement\text{Work} = \text{Force} \times \text{displacement}W=F⃗⋅d⃗W = \vec{F} \cdot \vec{d}
    • Vector Product (Cross Product):
      • Definition: Occurs when multiplying two vectors results in a vector product.
      • Example: Torque, defined as the cross product of position vector and force vector:             Torque=r⃗×F⃗\text{Torque} = \vec{r} \times \vec{F}τ⃗=(rFsin⁡(θ)) n^\vec{\tau} = (r F \sin(\theta))\,\hat{n}             where n^\hat{n} specifies the unit vector direction of torque.