Class 11 > Unit # 02: Kinematics > Addition of Vectors by Head to Tail rule

Class 11 Physics • Unit 2 Kinematics

Addition of Vectors by Head to Tail Method

Graphical vector addition rules, definition of resultant vector, and step-by-step graphical constructions for 2 and 3 vectors.

1. Fundamentals: Anatomy of a Vector & Resultant Vector

To perform graphical vector addition accurately, we must first understand the visual components of a directed line segment representing a vector:

1. The Head (Arrowhead)

The tip with the arrow showing the exact direction of the vector.

2. The Tail (Starting Point)

The starting point of the vector. The length of the line from tail to head represents the magnitude of the vector.

Resultant Vector ($\vec{R}$): A single vector which produces the same combined effect as produced by two or more vectors acting together on a body.

$$\vec{R} = \vec{A} + \vec{B} + \vec{C} + \dots$$

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2. Four Universal Steps of Head to Tail Rule

1. Draw First Vector: Choose a suitable scale and draw the first vector $\vec{A}$ maintaining its exact length and direction.
2. Join Next Vector (Head to Tail): Draw the second vector $\vec{B}$ such that its tail coincides with the head of the first vector $\vec{A}$. Repeat this for all subsequent vectors.
3. Draw Resultant Line: Join the tail of the first vector $\vec{A}$ to the head of the last vector with a straight line.
4. Direction of Resultant: Draw the arrowhead (head) of the resultant vector $\vec{R}$ meeting the head of the last vector.

Golden Rule of Graphical Vector Addition: Never alter the length or orientation (direction) of any vector when translating it across the grid. Head always joins to Tail; Head NEVER joins to Head or Tail to Tail during sequential addition!

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3. Practical Examples & Scale Constructions

Example 1: Adding Two Perpendicular Vectors

Suppose we want to add two vectors $\vec{A}$ and $\vec{B}$ graphically:

• Vector $\vec{A}$: Length $3\text{ cm}$ towards East ($0^\circ$).
• Vector $\vec{B}$: Length $4\text{ cm}$ towards North ($90^\circ$).

Step-by-Step Construction:

1. Draw $\vec{A} = 3\text{ cm}$ horizontally facing East.
2. At the head of $\vec{A}$, attach the tail of $\vec{B} = 4\text{ cm}$ vertically towards North.
3. Connect the tail of $\vec{A}$ to the head of $\vec{B}$ to form the resultant vector $\vec{R} = \vec{A} + \vec{B}$.

Verification with Scale & Protractor (D):

• Measured Length of $\vec{R}$: $5\text{ cm}$ (By Pythagorean theorem $R = \sqrt{3^2 + 4^2} = 5\text{ cm}$).
• Measured Angle ($\theta$): Place protractor origin at tail of $\vec{A}$ $\implies \theta = 53^\circ$ North of East.

Example 2: Adding Three Vectors

Consider three vectors defined with scale measurements:

• Vector $\vec{A}$: Length $3\text{ cm}$ at $45^\circ$ (North-East).
• Vector $\vec{B}$: Length $4\text{ cm}$ horizontally towards East.
• Vector $\vec{C}$: Length $3\text{ cm}$ at $45^\circ$ towards South-West (opposite to $\vec{A}$).

Step-by-Step Construction:

1. Draw $\vec{A}$ at $45^\circ$ with length $3\text{ cm}$.
2. Attach the tail of $\vec{B} = 4\text{ cm}$ (pointing East) to the head of $\vec{A}$.
3. Attach the tail of $\vec{C} = 3\text{ cm}$ (pointing $45^\circ$ South-West) to the head of $\vec{B}$.
4. Join the tail of first vector $\vec{A}$ to the head of last vector $\vec{C}$ with resultant arrow $\vec{R}$.

Resultant Verification:

Since $\vec{A}$ and $\vec{C}$ are equal in magnitude ($3\text{ cm}$) and opposite in direction, their effect cancels out completely. The resultant line will be perfectly horizontal facing East with a length equal to $\vec{B}$:

$$\text{Resultant Length } |\vec{R}| = 4\text{ cm} \quad (\text{Direction: East } / 0^\circ)$$

Summary Tip: Regardless of whether you add 2, 3, or 5 vectors, vector addition obeys the commutative law ($\vec{A} + \vec{B} = \vec{B} + \vec{A}$). Changing the sequence of adding vectors yields the exact same resultant vector magnitude and direction!